LOGIC, PROGRAMMING AND PROLOG (2ED) Ulf Nilsson and Jan Maluszy´ nski
c Copyright 2000, Ulf Nilsson and Jan Maluszy´ nski. The book may be downloaded and printed for personal use only provided that the text (1) is not altered in any way, and (2) is accompanied by this copyright notice. The book may also be copied and distributed in paper-form for non-profit use only. No other form of distribution or storage is permitted. In particular, it is not allowed to store and distribute the book electronically. This book was previously published by John Wiley & Sons Ltd. The book was originally published in 1990 with the second edition published in 1995. The copyright was reverted back to the authors in November 2000. For further information about updates and supplementary material please check out the book web-site at http://www.ida.liu.se/~ulfni/lpp or contact the authors at
[email protected] and
[email protected].
Contents
Preface
ix
I
1
Foundations
1 Preliminaries 1.1 Logic Formulas . . . . . . . . . . 1.2 Semantics of Formulas . . . . . . 1.3 Models and Logical Consequence 1.4 Logical Inference . . . . . . . . . 1.5 Substitutions . . . . . . . . . . . Exercises . . . . . . . . . . . . .
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3 3 7 10 13 14 16
2 Definite Logic Programs 2.1 Definite Clauses . . . . . . . . . 2.2 Definite Programs and Goals . 2.3 The Least Herbrand Model . . 2.4 Construction of Least Herbrand Exercises . . . . . . . . . . . .
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19 19 21 24 29 31
3 SLD-Resolution 3.1 Informal Introduction . . . . . 3.2 Unification . . . . . . . . . . . 3.3 SLD-Resolution . . . . . . . . . 3.4 Soundness of SLD-resolution . 3.5 Completeness of SLD-resolution 3.6 Proof Trees . . . . . . . . . . . Exercises . . . . . . . . . . . .
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33 33 37 43 48 51 53 57
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vi
Contents
4 Negation in Logic Programming 4.1 Negative Knowledge . . . . . . . . . . . 4.2 The Completed Program . . . . . . . . . 4.3 SLDNF-resolution for Definite Programs 4.4 General Logic Programs . . . . . . . . . 4.5 SLDNF-resolution for General Programs 4.6 Three-valued Completion . . . . . . . . 4.7 Well-founded Semantics . . . . . . . . . Exercises . . . . . . . . . . . . . . . . .
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59 59 61 65 67 70 75 77 84
5 Towards Prolog: Cut and Arithmetic 5.1 Cut: Pruning the SLD-tree . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Built-in Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87 87 93 97
II
99
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Programming in Logic
6 Logic and Databases 6.1 Relational Databases . . . . . . . . . . 6.2 Deductive Databases . . . . . . . . . . 6.3 Relational Algebra vs. Logic Programs 6.4 Logic as a Query-language . . . . . . . 6.5 Special Relations . . . . . . . . . . . . 6.6 Databases with Compound Terms . . Exercises . . . . . . . . . . . . . . . . 7 Programming with Recursive Data 7.1 Recursive Data Structures . . . . . 7.2 Lists . . . . . . . . . . . . . . . . . 7.3 Difference Lists . . . . . . . . . . . Exercises . . . . . . . . . . . . . .
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101 101 103 104 107 109 114 116
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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119 119 119 129 131
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8 Amalgamating Object- and Meta-language 8.1 What is a Meta-language? . . . . . . . . . . 8.2 Ground Representation . . . . . . . . . . . 8.3 Nonground Representation . . . . . . . . . . 8.4 The Built-in Predicate clause/2 . . . . . . . 8.5 The Built-in Predicates assert{a,z}/1 . . . . 8.6 The Built-in Predicate retract/1 . . . . . . . Exercises . . . . . . . . . . . . . . . . . . .
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135 135 136 141 143 144 146 146
9 Logic and Expert Systems 9.1 Expert Systems . . . . . . . . . . . . 9.2 Collecting Proofs . . . . . . . . . . . 9.3 Query-the-user . . . . . . . . . . . . 9.4 Fixing the Car (Extended Example) Exercises . . . . . . . . . . . . . . .
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149 149 153 154 155 161
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Contents
vii
10 Logic and Grammars 10.1 Context-free Grammars . . . . . . 10.2 Logic Grammars . . . . . . . . . . 10.3 Context-dependent Languages . . . 10.4 Definite Clause Grammars (DCGs) 10.5 Compilation of DCGs into Prolog . Exercises . . . . . . . . . . . . . .
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163 163 166 169 171 175 176
11 Searching in a State-space 11.1 State-spaces and State-transitions . . . . 11.2 Loop Detection . . . . . . . . . . . . . . 11.3 Water-jug Problem (Extended Example) 11.4 Blocks World (Extended Example) . . . 11.5 Alternative Search Strategies . . . . . . Exercises . . . . . . . . . . . . . . . . .
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179 179 181 182 183 185 186
III
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Alternative Logic Programming Schemes
189
12 Logic Programming and Concurrency 12.1 Algorithm = Logic + Control . . . . . 12.2 And-parallelism . . . . . . . . . . . . . 12.3 Producers and Consumers . . . . . . . 12.4 Don’t Care Nondeterminism . . . . . . 12.5 Concurrent Logic Programming . . . . Exercises . . . . . . . . . . . . . . . .
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191 191 193 194 196 196 202
13 Logic Programs with Equality 13.1 Equations and E-unification . 13.2 More on E-unification . . . . 13.3 Logic Programs with Equality Exercises . . . . . . . . . . .
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203 204 205 207 212
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14 Constraint Logic Programming 14.1 Logic Programming with Constraints 14.2 Declarative Semantics of CLP . . . . 14.3 Operational Semantics of CLP . . . 14.4 Examples of CLP-languages . . . . . Exercises . . . . . . . . . . . . . . .
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213 214 215 216 222 227
15 Query-answering in Deductive Databases 15.1 Naive Evaluation . . . . . . . . . . . . . . 15.2 Semi-naive Evaluation . . . . . . . . . . . 15.3 Magic Transformation . . . . . . . . . . . 15.4 Optimizations . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . .
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229 230 232 233 236 239
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viii
Contents
A Bibliographical Notes 241 A.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 A.2 Programming in Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 A.3 Alternative Logic Programming Schemes . . . . . . . . . . . . . . . . . 247 B Basic Set Theory 251 B.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 B.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 B.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 C Answers to Selected Exercises
253
Bibliography
263
Index
277
Preface
Since the first edition of this book the field of logic programming has developed and matured in many respects. This has been reflected by the large number of textbooks that appeared in that period. These books usually fall into one of the following three categories: • books which provide a theoretical basis for logic programming; • books which describe how to write programs in Prolog (sometimes even in particular Prolog systems); • books which describe alternative logic programming languages like constraint logic programming, deductive databases or concurrent logic programming.
Objectives The main objective of both editions of this textbook is to provide a uniform account of both the foundations of logic programming and simple programming techniques in the programming language Prolog. The discussion of the foundations also facilitates a systematic survey of variants of the logic programming scheme, like constraint logic programming, deductive databases or concurrent logic programming. This book is not primarily intended to be a theoretical handbook on logic programming. Nor is it intended to be a book on advanced Prolog programming or on constraint logic programming. For each of these topics there are more suitable books around. Because of the diversity of the field there is of course a risk that nothing substantial is said about anything. We have tried to compensate for this risk by limiting our attention to (what we think are) the most important areas of logic programming and by providing the interested reader with pointers containing suggestions for further reading. As a consequence of this: ix
x
Preface • the theoretical presentation is limited to well-established results and many of the most elaborate theorems are stated only with hints or pointers to their proofs; • most of the program examples are small programs whose prime aim is to illustrate the principal use of logic programming and to inspire the reader to apply similar techniques when writing “real” logic programs.
The objectives of the book have not changed since the first edition, but its content has been revised and updated to reflect the development of the field.
Prerequisites Like many other textbooks, this book emerged out of lecture notes which finally stabilized after several years of teaching. It has been used as introductory reading in the logic programming course for third year undergraduate students mainly from the computer science curriculum at Link¨ oping University. To take full benefit from the book, introductory courses in logic and discrete mathematics are recommended. Some basic knowledge in automata theory may be helpful but is not strictly necessary.
Organization The book is divided into three parts: • Foundations; • Programming in Logic; • Alternative Logic Programming Schemes. The first part deals with the logical aspects of logic programming and tries to provide a logical understanding of the programming language Prolog. Logic programs consist of logical formulas and computation is the process of deduction or proof construction. This makes logic programming fundamentally different from most other programming languages, largely a consequence of the fact that logic is considerably much older than electronic computers and not restricted to the view of computation associated with the Von Neumann machine. The main difference between logic programming and conventional programming languages is the declarative nature of logic. A program written in, for instance, Fortran can, in general, not be understood without taking operational considerations into account. That is, a Fortran program cannot be understood without knowing how it is going to be executed. In contrast to that, logic has no inherent concept of execution and logic formulas can be understood without any notion of evaluation or execution in mind. One of the most important aims of this book is to emphasize this distinction between logic programs and programs written in traditional programming languages. Chapter 1 contains a recapitulation of notions basic to logic in general. Readers who are already well acquainted with predicate logic can without problem omit this chapter. The chapter discusses concepts related both to model- and proof-theory of
Preface
xi
predicate logic including notions like language, interpretation, model, logical consequence, logical inference, soundness and completeness. The final section introduces the concept of substitution which is needed in subsequent chapters. Chapter 2 introduces the restricted language of definite programs and discusses the model-theoretic consequences of restricting the language. By considering only definite programs it suffices to limit attention to so-called Herbrand interpretations making the model-theoretic treatment of the language much simpler than for the case of full predicate logic. The operational semantics of definite programs is described in Chapter 3. The starting point is the notion of unification. A unification algorithm is provided and proved correct. Some of its properties are discussed. The unification algorithm is the basis for SLD-resolution which is the only inference rule needed for definite programs. Soundness and completeness of this rule are discussed. The use of negation in logic programming is discussed in Chapter 4. It introduces the negation-as-finite-failure rule used to implement negation in most Prolog systems and also provides a logical justification of the rule by extending the user’s program with additional axioms. Thereafter definite programs are generalized to general programs. The resulting proof-technique of this language is called SLDNF-resolution and is a result of combining SLD-resolution with the negation-as-finite-failure rule. Results concerning soundness of both the negation-as-finite-failure rule and SLDNF-resolution are discussed. Finally some alternative approaches based on three-valued logics are described to explain alternative views of negation in logic programming. The final chapter of Part I introduces two notions available in existing Prolog systems. Cut is introduced as a mechanism for reducing the overhead of Prolog computations. The main objective of this section is to illustrate the effect of cut and to point out cases when its use is motivated, and cases of misuse of cut. The conclusion is that cut should be used with great care and can often be avoided. For example, cut is not used in subsequent chapters, where many example programs are presented. The second section of Chapter 5 discusses the use of predefined arithmetic predicates in Prolog and provides a logical explanation for them. The second part of the book is devoted to some simple, but yet powerful, programming techniques in Prolog. The goal is not to study implementation-specific details of different Prolog systems nor is it our aim to develop real-size or highly optimized programs. The intention is rather to emphasize two basic principles which are important to appreciate before one starts considering writing “real” programs: • logic programs are used to describe relations, and • logic programs have both a declarative and an operational meaning. In order to write good programs it is important to keep both aspects in mind. Part II of the book is divided into several chapters which relate logic programming to different fields of computer science while trying to emphasize these two points. Chapter 6 describes logic programming from a database point of view. It is shown how logic programs can be used, in a coherent way, as a framework for representing relational databases and for retrieving information out of them. The chapter also contains some extensions to traditional databases. For instance, the ability to define infinite relations and the use of structured data.
xii
Preface
Chapter 7 demonstrates techniques for defining relations on recursive data-structures, in particular on lists. The objective is to study how recursive data-structures give rise to recursive programs which can be defined in a uniform way by means of inductive definitions. The second part of the chapter presents an alternative representation of lists and discusses advantages and disadvantages of this new representation. Chapter 8 introduces the notion of meta- and object-language and illustrates how to use logic programs for describing SLD-resolution. The ability to do this in a simple way facilitates some very powerful programming techniques. The chapter also introduces some (controversial) built-in predicates available in most Prolog implementations. Chapter 9 is a continuation of Chapter 8. It demonstrates how to extend an interpreter from Chapter 8 into a simple expert-system shell. The resulting program can be used as a starting point for developing a full-scale expert system. Historically one of the main objectives for implementing Prolog was its application for natural language processing. Chapter 10 shows how to describe grammars in Prolog, starting from context-free grammars. Thereafter larger classes of languages are considered. The last two sections introduce the notion of Definite Clause Grammars (DCGs) commonly used for describing both natural and artificial languages in Prolog. The last chapter of Part II elaborates on results from Chapter 6. The chapter demonstrates simple techniques for solving search-problems in state-transition graphs and raises some of the difficulties which are inherently associated with such problems. The final part of the book gives a brief introduction to some extensions of the logic programming paradigm, which are still subject of active research. Chapter 12 describes a class of languages commonly called concurrent logic programming languages. The underlying execution model of these languages is based on concurrent execution. It allows therefore for applications of logic programming for description of concurrent processes. The presentation concentrates on the characteristic principles of this class of languages, in particular on the mechanisms used to enforce synchronization between parallel processes and the notion of don’t care nondeterminism. Chapter 13 discusses an approach to integration of logic programming with functional programming based on the use of equations. The notion of E-unification (unification modulo a set E of equations) is introduced and properties of E-unification algorithms are discussed. Finally it is shown how to generalize the notion of SLDresolution to incorporate E-unification instead of “ordinary” unification. Chapter 14 concerns the use of constraints in logic programming. The constraint logic programming scheme has attracted a great many people because of its generality, elegance and expressive power. A rigorous semantical framework is briefly described. The main ideas are illustrated using examples from several constraint domains. The final chapter of Part III concerns the optimization of queries to deductive databases. The chapter provides an alternative to SLD-resolution as the inference mechanism in a query-answering system and discusses the principal idea of several optimizations described in the literature. In addition the book contains three appendices. The first of them provides bibliographical remarks to most of the chapters of the book including suggestions for further reading. The second appendix contains a brief account of set theoretic notions used throughout the book and the final appendix contains solutions and hints for some of the exercises which are available in the main text.
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What is new in the second edition? The second edition of the book contains one new chapter on query optimization in deductive databases (Chapter 15). Three chapters have also been substantially revised: The presentation of unification in Chapter 3 has been modified to facilitate better integration with Chapters 13 (equational logic programming) and 14 (constraint logic programming). To simplify the presentation of constraint logic programming, Chapter 3 also introduces the notion of derivation trees. Secondly, chapter 4 on negation has been completely revised. In particular, the definition of SLDNF-resolution has been improved and two new sections have been added covering alternative approaches to negation — three-valued completion and well-founded semantics. Finally, Chapter 14 has been substantially extended providing the theoretical foundation of the constraint logic programming scheme and several examples of constraint logic programming languages. Most of the remaining chapters have undergone minor modifications; new examples and exercises have been included, the bibliographical remarks have been updated and an appendix on basic set theory has been added.
Acknowledgements The authors would like to thank a number of persons for their involvement in the course of writing the first and second edition of this book. In particular, Roland Bol, Staffan Bonnier, Lars Degerstedt, Wlodzimierz Drabent and all other members of the Logic Programming Laboratory. We are also indebted to students, who lived through draft versions of the book and provided invaluable feedback. Thanks are also due to Gu Xinli, Jalal Maleki, Mirka Milkowska, Simin Nadjm-Tehrani, Torbj¨orn N¨ aslund and Linda Smith who devoted much of their time reading parts of the manuscript. Needless to say, the remaining flaws are to be attributed to the authors. Our deepest gratitude also to Roslyn Meredith and Rosemary Altoft at John Wiley, and the anonymous referees whose comments influenced the final structure and contents of both editions of the book. Finally we should mention that the material presented in this book is closely related to our research interests. We gratefully acknowledge the financial support of our research projects by the Swedish Research Council for Engineering Sciences (TFR) and by Link¨ oping University.
Link¨oping, Sweden June 1995
Ulf Nilsson ´ski Jan Maluszyn
xiv
Preface
PART I FOUNDATIONS
1
Chapter 1 Preliminaries
1.1
Logic Formulas
When describing some state of affairs in the real world we often use declarative1 sentences like: (i ) “Every mother loves her children” (ii ) “Mary is a mother and Tom is Mary’s child” By applying some general rules of reasoning such descriptions can be used to draw new conclusions. For example, knowing (i ) and (ii ) it is possible to conclude that: (iii ) “Mary loves Tom” A closer inspection reveals that (i ) and (ii ) describe some universe of persons and some relations between these individuals — like “. . . is a mother”, “. . . is a child of . . . ” or the relation “. . . loves . . . ” — which may or may not hold between the persons.2 This example reflects the principal idea of logic programming — to describe possibly infinite relations on objects and to apply the programming system in order to draw conclusions like (iii ). For a computer to deal with sentences like (i )–(iii ) the syntax of the sentences must be precisely defined. What is even more important, the rules of reasoning — like the one 1 The notion of declarative sentence has its roots in linguistics. A declarative sentence is a complete expression of natural language which is either true or false, as opposed to e.g. imperative or interrogative sentences (commands and questions). Only declarative sentences can be expressed in predicate logic. 2 Some people would probably argue that “being a mother” is not a relation but rather a property. However, for the sake of uniformity properties will be called relations and so will statements which relate more than two objects (like “. . . is the sum of . . . and . . . ”).
3
4
Chapter 1: Preliminaries
which permits inferring (iii ) from (i ) and (ii ) — must be carefully formalized. Such problems have been studied in the field of mathematical logic. This chapter surveys basic logical concepts that are used later on in the book to relate logic programming and logic. (For basic set theoretic notions see Appendix B.) The first concept considered is that of logic formulas which provide a formalized syntax for writing sentences like (i )–(iii ). Such sentences refer to individuals in some world and to relations between those individuals. Therefore the starting point is an assumption about the alphabet of the language. It must include: • symbols for denoting individuals (e.g. the symbol tom may be used to denote the person Tom of our example). Such symbols will be called constants; • symbols for denoting relations (loves , mother , child of ). Such symbols are called predicate symbols. Every predicate symbol has an associated natural number, called its arity. The relation named by an n-ary predicate symbol is a set of n-tuples of individuals; in the example above the predicate symbol loves denotes a set of pairs of persons, including the pair Mary and Tom, denoted by the constants mary and tom. With the alphabet of constants, predicate symbols and some auxiliary characters, sentences of natural language like “Mary loves Tom” can be formalized as formulas like loves(mary, tom). The formal language should also provide the possibility of expressing sentences like (i ) which refers to all elements of the described “world”. This sentence says that “for all individuals X and Y, if X is a mother and Y is a child of X then X loves Y”. For this purpose, the language of logic introduces the symbol of universal quantifier “∀ ” ( to be read “for every” or “for all”) and the alphabet of variables. A variable is a symbol that refers to an unspecified individual, like X and Y above. Now the sentences (i )–(iii ) can be formalized accordingly: ∀X (∀ Y ((mother (X) ∧ child of (Y, X)) ⊃ loves(X, Y ))) mother (mary) ∧ child of (tom, mary) loves(mary, tom)
(1) (2) (3)
The symbols “∧” and “⊃” are examples of logical connectives which are used to combine logic formulas — “∧” reads “and” and is called conjunction whereas “⊃” is called implication and corresponds to the “if-then” construction above. Parentheses are used to disambiguate the language. Another connective which will be used frequently is that for expressing negation. It is denoted by “¬” (with reading “not”). For example the sentence “Tom does not love Mary” can be formalized as the formula: ¬loves(tom, mary) In what follows the symbol “∃” is also sometimes used. It is called the existential quantifier and reads “there exists”. The existential quantifier makes it possible to express the fact that, in the world under consideration, there exists at least one individual
1.1 Logic Formulas
5
which is in a certain relation with some other individuals. For example the sentence “Mary has a child” can be formalized as the formula: ∃X child of (X, mary) On occasion the logical connectives “∨” and “↔” are used. They formalize the connectives “or” and “if and only if” (“iff”). So far individuals have been represented only by constants. However it is often the case that in the world under consideration, some “individuals” are “composed objects”. For instance, in some world it may be necessary to discuss relations between families as well as relations between persons. In this case it would be desirable to refer to a given family by a construction composed of the constants identifying the members of the family (actually what is needed is a function that constructs a family from its members). The language of logic offers means of solving this problem. It is assumed that its alphabet contains symbols called functors that represent functions over object domains. Every functor has assigned a natural number called its arity, which determines the number of arguments of the function. The constants can be seen as 0-ary functors. Assume now that there is a ternary 3 functor family, a binary functor child and a constant none. The family consisting of the parents Bill and Mary and children Tom and Alice can now be represented by the construction: family(bill , mary, child (tom, child (alice, none))) Such a construction is called a compound term. The above informal discussion based on examples of simple declarative sentences gives motivation for introducing basic constructs of the language of symbolic logic. The kind of logic used here is called predicate logic. Next a formal definition of this language is given. For the moment we specify only the form of allowed sentences, while the meaning of the language will be discussed separately. Thus the definition covers only the syntax of the language separated from its semantics. From the syntactic point of view logic formulas are finite sequences of symbols such as variables, functors and predicate symbols. There are infinitely many of them and therefore the symbols are usually represented by finite strings of primitive characters. The representation employed in this book usually conforms to that specified in the ISO standard of the programming language Prolog (1995). Thus, the alphabet of the language of predicate logic consists of the following classes of symbols: • variables which will be written as alphanumeric identifiers beginning with capital letters (sometimes subscriped). Examples of variables are X, Xs, Y, X7, . . .; • constants which are numerals or alphanumeric identifiers beginning with lowercase letters. Examples of constants are x, alf , none, 17, . . .; • functors which are alphanumeric identifiers beginning with lower-case letters and with an associated arity > 0. To emphasize the arity n of a functor f it is sometimes written in the form f /n; 3 Usually the terms nullary, unary, binary and ternary are used instead of 0-ary, 1-ary, 2-ary and 3-ary.
6
Chapter 1: Preliminaries • predicate symbols which are usually alphanumeric identifiers starting with lowercase letters and with an associated arity ≥ 0. The notation p/n is used also for predicate symbols; • logical connectives which are ∧ (conjunction), ¬ (negation), ↔ (logical equivalence), ⊃ (implication) and ∨ (disjunction); • quantifiers — ∀ (universal) and ∃ (existential); • auxiliary symbols like parentheses and commas.
No syntactic distinction will be imposed between constants, functors and predicate symbols. However, as a notational convention we use a, b, c, . . . (with or without adornments) to denote constants and X, Y, Z, . . . to denote variables. Functors are denoted f, g, h, . . . and p, q, r, . . . are used to denote predicate symbols. Constants are sometimes viewed as nullary functors. Notice also that the sets of functors and predicate symbols may contain identical identifiers with different arities. Sentences of natural language consist of words where objects of the described world are represented by nouns. In the formalized language of predicate logic objects will be represented by strings called terms whose syntax is defined as follows: Definition 1.1 (Terms) The set T of terms over a given alphabet A is the smallest set such that: • any constant in A is in T ; • any variable in A is in T ; • if f /n is a functor in A and t1 , . . . , tn ∈ T then f (t1 , . . . , tn ) ∈ T . In this book terms are typically denoted by s and t. In natural language only certain combinations of words are meaningful sentences. The counterpart of sentences in predicate logic are special constructs built from terms. These are called formulas or well-formed formulas (wff ) and their syntax is defined as follows: Definition 1.2 (Formulas) Let T be the set of terms over the alphabet A. The set F of wff (with respect to A) is the smallest set such that: • if p/n is a predicate symbol in A and t1 , . . . , tn ∈ T then p(t1 , . . . , tn ) ∈ F; • if F and G ∈ F then so are (¬F ), (F ∧ G), (F ∨ G), (F ⊃ G) and (F ↔ G); • if F ∈ F and X is a variable in A then (∀XF ) and (∃XF ) ∈ F. Formulas of the form p(t1 , . . . , tn ) are called atomic formulas (or simply atoms). In order to adopt a syntax similar to that of Prolog, formulas in the form (F ⊃ G) are instead written in the form (G ← F ). To simplify the notation parentheses will be removed whenever possible. To avoid ambiguity it will be assumed that the connectives
1.2 Semantics of Formulas
7
have a binding-order where ¬, ∀ and ∃ bind stronger than ∨, which in turn binds stronger than ∧ followed by ⊃ (i.e. ←) and finally ↔. Thus (a ← ((¬b) ∧ c)) will be simplified into a ← ¬b ∧ c. Sometimes binary functors and predicate symbols are written in infix notation (e.g. 2 ≤ 3). Let F be a formula. An occurrence of the variable X in F is said to be bound either if the occurrence follows directly after a quantifier or if it appears inside the subformula which follows directly after “∀X” or “∃X”. Otherwise the occurrence is said to be free. A formula with no free occurrences of variables is said to be closed. A formula/term which contains no variables is called ground. Let X1 , . . . , Xn be all variables that occur free in a formula F . The closed formula of the form ∀ X1 (. . . (∀ Xn F ) . . .) is called the universal closure of F and is denoted ∀ F . Similarly, ∃ F is called the existential closure of F and denotes the formula F closed under existential quantification.
1.2
Semantics of Formulas
The previous section introduced the language of formulas as a formalization of a class of declarative statements of natural language. Such sentences refer to some “world” and may be true or false in this world. The meaning of a logic formula is also defined relative to an “abstract world” called an (algebraic) structure and is also either true or false. In other words, to define the meaning of formulas, a formal connection between the language and a structure must be established. This section discusses the notions underlying this idea. As stated above declarative statements refer to individuals, and concern relations and functions on individuals. Thus the mathematical abstraction of the “world”, called a structure, is a nonempty set of individuals (called the domain) with a number of relations and functions defined on this domain. For example the structure referred to by the sentences (i )–(iii ) may be an abstraction of the world shown in Figure 1.1. Its domain consists of three individuals — Mary, John and Tom. Moreover, three relations will be considered on this set: a unary relation, “. . . is a mother”, and two binary relations, “. . . is a child of . . . ” and “. . . loves . . . ”. For the sake of simplicity it is assumed that there are no functions in the structure. The building blocks of the language of formulas are constants, functors and predicate symbols. The link between the language and the structure is established as follows: Definition 1.3 (Interpretation) An interpretation = of an alphabet A is a nonempty domain D (sometimes denoted |=|) and a mapping that associates: • each constant c ∈ A with an element c= ∈ D; • each n-ary functor f ∈ A with a function f= : Dn → D; • each n-ary predicate symbol p ∈ A with a relation p= ⊆ D × · · · × D . | {z } n
The interpretation of constants, functors and predicate symbols provides a basis for assigning truth values to formulas of the language. The meaning of a formula will be
8
Chapter 1: Preliminaries
Mary
Tom
John
Figure 1.1: A family structure
defined as a function on meanings of its components. First the meaning of terms will be defined since they are components of formulas. Since terms may contain variables the auxiliary notion of valuation is needed. A valuation ϕ is a mapping from variables of the alphabet to the domain of an interpretation. Thus, it is a function which assigns objects of an interpretation to variables of the language. By the notation ϕ[X 7→ t] we denote the valuation which is identical to ϕ except that ϕ[X 7→ t] maps X to t. Definition 1.4 (Semantics of terms) Let = be an interpretation, ϕ a valuation and t a term. Then the meaning ϕ= (t) of t is an element in |=| defined as follows: • if t is a constant c then ϕ= (t) := c= ; • if t is a variable X then ϕ= (t) := ϕ(X); • if t is of the form f (t1 , . . . , tn ), then ϕ= (t) := f= (ϕ= (t1 ), . . . , ϕ= (tn )). Notice that the meaning of a compound term is obtained by applying the function denoted by its main functor to the meanings of its principal subterms, which are obtained by recursive application of this definition. Example 1.5 Consider a language which includes the constant zero, the unary functor s and the binary functor plus. Assume that the domain of = is the set of the natural numbers (N ) and that: zero=
:= 0
1.2 Semantics of Formulas
9 s= (x) := 1 + x plus= (x, y) := x + y
That is, zero denotes the natural number 0, s denotes the successor function and plus denotes the addition function. For the interpretation = and a valuation ϕ such that ϕ(X) := 0 the meaning of the term plus(s(zero), X) is obtained as follows: ϕ= (plus(s(zero), X)) = = = =
ϕ= (s(zero)) + ϕ= (X) (1 + ϕ= (zero)) + ϕ(X) (1 + 0) + 0 1
The meaning of a formula is a truth value. The meaning depends on the components of the formula which are either (sub-) formulas or terms. As a consequence the meanings of formulas also rely on valuations. In the following definition the notation = |=ϕ Q is used as a shorthand for the statement “Q is true with respect to = and ϕ” and = 6|=ϕ Q is to be read “Q is false w.r.t. = and ϕ”. Definition 1.6 (Semantics of wff ’s) Let = be an interpretation, ϕ a valuation and Q a formula. The meaning of Q w.r.t. = and ϕ is defined as follows: • = |=ϕ p(t1 , . . . , tn ) iff hϕ= (t1 ), . . . , ϕ= (tn )i ∈ p= ; • = |=ϕ (¬F ) iff = 6|=ϕ F ; • = |=ϕ (F ∧ G) iff = |=ϕ F and = |=ϕ G; • = |=ϕ (F ∨ G) iff = |=ϕ F or = |=ϕ G (or both); • = |=ϕ (F ⊃ G) iff = |=ϕ G whenever = |=ϕ F ; • = |=ϕ (F ↔ G) iff = |=ϕ (F ⊃ G) and = |=ϕ (G ⊃ F ); • = |=ϕ (∀XF ) iff = |=ϕ[X7→t] F for every t ∈ |=|; • = |=ϕ (∃XF ) iff = |=ϕ[X7→t] F for some t ∈ |=|.
The semantics of formulas as defined above relies on the auxiliary concept of valuation that associates variables of the formula with elements of the domain of the interpretation. It is easy to see that the truth value of a closed formula depends only on the interpretation. It is therefore common practice in logic programming to consider all formulas as being implicitly universally quantified. That is, whenever there are free occurrences of variables in a formula its universal closure is considered instead. Since the valuation is of no importance for closed formulas it will be omitted when considering the meaning of such formulas.
10
Chapter 1: Preliminaries
Example 1.7 Consider Example 1.5 again. Assume that the language contains also a unary predicate symbol p and that: p= := {h1i, h3i, h5i, h7i, . . .} Then the meaning of the formula p(zero) ∧ p(s(zero)) in the interpretation = is determined as follows: = |= p(zero) ∧ p(s(zero)) iff iff iff iff
= |= p(zero) and = |= p(s(zero)) hϕ= (zero)i ∈ p= and hϕ= (s(zero))i ∈ p= hϕ= (zero)i ∈ p= and h1 + ϕ= (zero)i ∈ p= h0i ∈ p= and h1i ∈ p=
Now h1i ∈ p= but h0i 6∈ p= so the whole formula is false in =. Example 1.8 Consider the interpretation = that assigns: • the persons Tom, John and Mary of the structure in Figure 1.1 to the constants tom, john and mary; • the relations “. . . is a mother”, “. . . is a child of . . . ” and “. . . loves . . . ” of the structure in Figure 1.1 to the predicate symbols mother/1, child of /2 and loves/2. Using the definition above it is easy to show that the meaning of the formula: ∀X ∃ Y loves(X, Y ) is false in = (since Tom does not love anyone), while the meaning of formula: ∃X ∀ Y ¬loves(Y, X) is true in = (since Mary is not loved by anyone).
1.3
Models and Logical Consequence
The motivation for introducing the language of formulas was to give a tool for describing “worlds” — that is, algebraic structures. Given a set of closed formulas P and an interpretation = it is natural to ask whether the formulas of P give a proper account of this world. This is the case if all formulas of P are true in =. Definition 1.9 (Model) An interpretation = is said to be a model of P iff every formula of P is true in =. Clearly P has infinitely many interpretations. However, it may happen that none of them is a model of P . A trivial example is any P that includes the formula (F ∧ ¬F ) where F is an arbitrary (closed) formula. Such sets of formulas are called unsatisfiable. When using formulas for describing “worlds” it is necessary to make sure that every description produced is satisfiable (that is, has at least one model), and in particular that the world being described is a model of P . Generally, a satisfiable set of formulas has (infinitely) many models. This means that the formulas which properly describe a particular “world” of interest at the same time describe many other worlds.
1.3 Models and Logical Consequence
11
C B A
Figure 1.2: An alternative structure
Example 1.10 Figure 1.2 shows another structure which can be used as a model of the formulas (1) and (2) of Section 1.1 which were originally used to describe the world of Figure 1.1. In order for the structure to be a model the constants tom, john and mary are interpreted as the boxes ‘A’, ‘B’ and ‘C’ respectively — the predicate symbols loves, child of and mother are interpreted as the relations “. . . is above . . . ”, “. . . is below . . . ” and “. . . is on top”. Our intention is to use the description of the world of interest to obtain more information about this world. This new information is to be represented by new formulas not explicitly included in the original description. An example is the formula (3) of Section 1.1 which is obtained from (1) and (2). In other words, for a given set P of formulas other formulas (say F ) which are also true in the world described by P are searched for. Unfortunately, P itself has many models and does not uniquely identify the “intended model” which was described by P . Therefore it must be required that F is true in every model of P to guarantee that it is also true in the particular world of interest. This leads to the fundamental concept of logical consequence. Definition 1.11 (Logical consequence) Let P be a set of closed formulas. A closed formula F is called a logical consequence of P (denoted P |= F ) iff F is true in every model of P . Example 1.12 To illustrate this notion by an example it is shown that (3) is a logical consequence of (1) and (2). Let = be an arbitrary interpretation. If = is a model of (1) and (2) then: = |= ∀X(∀ Y ((mother (X) ∧ child of (Y, X)) ⊃ loves(X, Y ))) = |= mother (mary) ∧ child of (tom, mary)
(4) (5)
For (4) to be true it is necessary that: = |=ϕ mother (X) ∧ child of (Y, X) ⊃ loves(X, Y )
(6)
for any valuation ϕ — specifically for ϕ(X) = mary = and ϕ(Y ) = tom = . However, since these individuals are denoted by the constants mary and tom it must also hold that: = |= mother (mary) ∧ child of (tom, mary) ⊃ loves(mary, tom)
(7)
Finally, for this to hold it follows that loves(mary, tom) must be true in = (by Definition 1.6 and since (5) holds by assumption). Hence, any model of (1) and (2) is also a model of (3).
12
Chapter 1: Preliminaries
This example shows that it may be rather difficult to prove that a formula is a logical consequence of a set of formulas. The reason is that one has to use the semantics of the language of formulas and to deal with all models of the formulas. One possible way to prove P |= F is to show that ¬F is false in every model of P , or put alternatively, that the set of formulas P ∪ {¬F } is unsatisfiable (has no model). The proof of the following proposition is left as an exercise. Proposition 1.13 (Unsatisfiability) Let P be a set of closed formulas and F a closed formula. Then P |= F iff P ∪ {¬F } is unsatisfiable. It is often straightforward to show that a formula F is not a logical consequence of the set P of formulas. For this, it suffices to give a model of P which is not a model of F . Example 1.14 Let P be the formulas: ∀X(r(X) ⊃ (p(X) ∨ q(X)))
(8)
r(a) ∧ r(b)
(9)
To prove that p(a) is not a logical consequence of P it suffices to consider an interpretation = where |=| is the set consisting of the two persons “Adam” and “Eve” and where: a= := Adam b= := Eve p= := {hEvei} q= := {hAdami} r= := {hAdami, hEvei}
% the property of being female % the property of being male % the property of being a person
Clearly, (8) is true in = since “any person is either female or male”. Similarly (9) is true since “both Adam and Eve are persons”. However, p(a) is false in = since Adam is not a female. Another important concept based on the semantics of formulas is the notion of logical equivalence. Definition 1.15 (Logical equivalence) Two formulas F and G are said to be logically equivalent (denoted F ≡ G) iff F and G have the same truth value for all interpretations = and valuations ϕ. Next a number of well-known facts concerning equivalences of formulas are given. Let F and G be arbitrary formulas and H(X) a formula with zero or more free occurrences of X. Then: ¬¬F F ⊃G F ⊃G F ↔G ¬(F ∨ G) ¬(F ∧ G) ¬∀XH(X) ¬∃XH(X)
≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡
F ¬F ∨ G ¬G ⊃ ¬F (F ⊃ G) ∧ (G ⊃ F ) ¬F ∧ ¬G ¬F ∨ ¬G ∃X¬H(X) ∀X¬H(X)
DeMorgan’s DeMorgan’s DeMorgan’s DeMorgan’s
law law law law
1.4 Logical Inference
13
and if there are no free occurrences of X in F then: ∀X(F ∨ H(X)) ≡ F ∨ ∀XH(X) Proofs of these equivalences are left as an exercise to the reader.
1.4
Logical Inference
In Section 1.1 the sentence (iii ) was obtained by reasoning about the sentences (i ) and (ii ). The language was then formalized and the sentences were expressed as the logical formulas (1), (2) and (3). With this formalization, reasoning can be seen as a process of manipulation of formulas, which from a given set of formulas, like (1) and (2), called the premises, produces a new formula called the conclusion, for instance (3). One of the objectives of the symbolic logic is to formalize “reasoning principles” as formal re-write rules that can be used to generate new formulas from given ones. These rules are called inference rules. It is required that the inference rules correspond to correct ways of reasoning — whenever the premises are true in any world under consideration, any conclusion obtained by application of an inference rule should also be true in this world. In other words it is required that the inference rules produce only logical consequences of the premises to which they can be applied. An inference rule satisfying this requirement is said to be sound. Among well-known inference rules of predicate logic the following are frequently used: • Modus ponens or elimination rule for implication: This rule says that whenever formulas of the form F and (F ⊃ G) belong to or are concluded from a set of premises, G can be inferred. This rule is often presented as follows: F
F ⊃G G
(⊃ E)
• Elimination rule for universal quantifier: This rule says that whenever a formula of the form (∀XF ) belongs to or is concluded from the premises a new formula can be concluded by replacing all free occurrences of X in F by some term t which is free for X (that is, all variables in t remain free when X is replaced by t: for details see e.g. van Dalen (1983) page 68). This rule is often presented as follows: ∀XF (X) (∀ E) F (t) • Introduction rule for conjunction: This rule states that if formulas F and G belong to or are concluded from the premises then the conclusion F ∧ G can be inferred. This is often stated as follows: F G F ∧G
(∧I)
Soundness of these rules can be proved directly from the definition of the semantics of the language of formulas.
14
Chapter 1: Preliminaries Their use can be illustrated by considering the example above. The premises are: ∀X (∀ Y (mother (X) ∧ child of (Y, X) ⊃ loves(X, Y ))) mother (mary) ∧ child of (tom, mary)
(10) (11)
Elimination of the universal quantifier in (10) yields: ∀ Y (mother (mary) ∧ child of (Y, mary) ⊃ loves(mary, Y ))
(12)
Elimination of the universal quantifier in (12) yields: mother (mary) ∧ child of (tom, mary) ⊃ loves(mary, tom)
(13)
Finally modus ponens applied to (11) and (13) yields: loves(mary, tom)
(14)
Thus the conclusion (14) has been produced in a formal way by application of the inference rules. The example illustrates the concept of derivability. As observed, (14) is obtained from (10) and (11) not directly, but in a number of inference steps, each of them adding a new formula to the initial set of premises. Any formula F that can be obtained in that way from a given set P of premises is said to be derivable from P . This is denoted by P ` F . If the inference rules are sound it follows that whenever P ` F , then P |= F . That is, whatever can be derived from P is also a logical consequence of P . An important question related to the use of inference rules is the problem of whether all logical consequences of an arbitrary set of premises P can also be derived from P . In this case the set of inference rules is said to be complete. Definition 1.16 (Soundness and Completeness) A set of inference rules are said to be sound if, for every set of closed formulas P and every closed formula F , whenever P ` F it holds that P |= F . The inference rules are complete if P ` F whenever P |= F . A set of premises is said to be inconsistent if any formula can be derived from the set. Inconsistency is the proof-theoretic counterpart of unsatisfiability, and when the inference system is both sound and complete the two are frequently used as synonyms.
1.5
Substitutions
The chapter is concluded with a brief discussion on substitutions — a concept fundamental to forthcoming chapters. Formally a substitution is a mapping from variables of a given alphabet to terms in this alphabet. The following syntactic definition is often used instead: Definition 1.17 (Substitutions) A substitution is a finite set of pairs of terms {X1 /t1 , . . . , Xn /tn } where each ti is a term and each Xi a variable such that Xi 6= ti and Xi 6= Xj if i 6= j. The empty substitution is denoted .
1.5 Substitutions
15
The application Xθ of a substitution θ to a variable X is defined as follows: t if X/t ∈ θ. Xθ := X otherwise In what follows let Dom({X1 /t1 , . . . , Xn /tn }) denote the set {X1 , . . . , Xn }. Also let Range({X1 /t1 , . . . , Xn /tn }) be the set of all variables in t1 , . . . , tn . Thus, for variables not included in Dom(θ), θ behaves as the identity mapping. It is natural to extend the domain of substitutions to include also terms and formulas. In other words, it is possible to apply a substitution to an arbitrary term or formula in the following way: Definition 1.18 (Application) Let θ be a substitution {X1 /t1 , . . . , Xn /tn } and E a term or a formula. The application Eθ of θ to E is the term/formula obtained by simultaneously replacing ti for every free occurrence of Xi in E (1 ≤ i ≤ n). Eθ is called an instance of E. Example 1.19 p(f (X, Z), f (Y, a)){X/a, Y /Z, W/b} = p(f (a, Z), f (Z, a)) p(X, Y ){X/f (Y ), Y /b} = p(f (Y ), b)
It is also possible to compose substitutions: Definition 1.20 (Composition) Let θ and σ be two substitutions: θ σ
:= {X1 /s1 , . . . , Xm /sm } := {Y1 /t1 , . . . , Yn /tn }
The composition θσ of θ and σ is obtained from the set: {X1 /s1 σ, . . . , Xm /sm σ, Y1 /t1 , . . . , Yn /tn } by removing all Xi /si σ for which Xi = si σ (1 ≤ i ≤ m) and by removing those Yj /tj for which Yj ∈ {X1 , . . . , Xm } (1 ≤ j ≤ n). It is left as an exercise to prove that the above syntactic definition of composition actually coincides with function composition (see exercise 1.13). Example 1.21 {X/f (Z), Y /W }{X/a, Z/a, W/Y } = {X/f (a), Z/a, W/Y } A kind of substitution that will be of special interest are the so-called idempotent substitutions: Definition 1.22 (Idempotent substitution) A substitution θ is said to be idempotent iff θ = θθ.
16
Chapter 1: Preliminaries
It can be shown that a substitution θ is idempotent iff Dom(θ) ∩ Range(θ) = ?. The proof of this is left as an exercise and so are the proofs of the following properties: Proposition 1.23 (Properties of substitutions) Let θ, σ and γ be substitutions and let E be a term or a formula. Then: • E(θσ) = (Eθ)σ • (θσ)γ = θ(σγ) • θ = θ = θ Notice that composition of substitutions is not commutative as illustrated by the following example: {X/f (Y )}{Y /a} = {X/f (a), Y /a} 6= {Y /a}{X/f (Y )} = {Y /a, X/f (Y )}
Exercises 1.1 Formalize the following sentences of natural language as formulas of predicate logic: a) b) c) d) e) f)
Every natural number has a successor. Nothing is better than taking a nap. There is no such thing as negative integers. The names have been changed to protect the innocent. Logic plays an important role in all areas of computer science. The renter of a car pays the deductible in case of an accident.
1.2 Formalize the following sentences of natural language into predicate logic: a) b) c)
A bronze medal is better than nothing. Nothing is better than a gold medal. A bronze medal is better than a gold medal.
1.3 Prove Proposition 1.13. 1.4 Prove the equivalences in connection with Definition 1.15. 1.5 Let F := ∀X ∃ Y p(X, Y ) and G := ∃ Y ∀Xp(X, Y ). State for each of the following four formulas whether it is satisfiable or not. If it is, give a model with the natural numbers as domain, if it is not, explain why. (F ∧ G) (F ∧ ¬G)
(¬F ∧ ¬G)
(¬F ∧ G)
1.6 Let F and G be closed formulas. Show that F ≡ G iff {F } |= G and {G} |= F . 1.7 Show that P is unsatisfiable iff there is some closed formula F such that P |= F and P |= ¬F .
Exercises
17
1.8 Show that the following three formulas are satisfiable only if the interpretation has an infinite domain ∀X¬p(X, X) ∀X∀Y ∀Z(p(X, Y ) ∧ p(Y, Z) ⊃ p(X, Z)) ∀X∃Y p(X, Y ) 1.9 Let F be a formula and θ a substitution. Show that ∀F |= ∀(F θ). 1.10 Let P1 , P2 and P3 be sets of closed formulas. Redefine |= in such a way that P1 |= P2 iff every formula in P2 is a logical consequence of P1 . Then show that |= is transitive — that is, if P1 |= P2 and P2 |= P3 then P1 |= P3 . 1.11 Let P1 and P2 be sets of closed formulas. Show that if P1 ⊆ P2 and P1 |= F then P2 |= F . 1.12 Prove Proposition 1.23. 1.13 Let θ and σ be substitutions. Show that the composition θσ is equivalent to function composition of the mappings denoted by θ and σ. 1.14 Show that a substitution θ is idempotent iff Dom(θ) ∩ Range(θ) = ?. 1.15 Which of the following statements are true? • if σθ = δθ then σ = δ • if θσ = θδ then σ = δ • if σ = δ then σθ = δθ
18
Chapter 1: Preliminaries
Chapter 2 Definite Logic Programs
2.1
Definite Clauses
The idea of logic programming is to use a computer for drawing conclusions from declarative descriptions. Such descriptions — called logic programs — consist of finite sets of logic formulas. Thus, the idea has its roots in the research on automatic theorem proving. However, the transition from experimental theorem proving to applied logic programming requires improved efficiency of the system. This is achieved by introducing restrictions on the language of formulas — restrictions that make it possible to use the relatively simple and powerful inference rule called the SLD-resolution principle. This chapter introduces a restricted language of definite logic programs and in the next chapter their computational principles are discussed. In subsequent chapters a more unrestrictive language of so-called general programs is introduced. In this way the foundations of the programming language Prolog are presented. To start with, attention will be restricted to a special type of declarative sentences of natural language that describe positive facts and rules. A sentence of this type either states that a relation holds between individuals (in case of a fact), or that a relation holds between individuals provided that some other relations hold (in case of a rule). For example, consider the sentences: (i ) “Tom is John’s child” (ii ) “Ann is Tom’s child” (iii ) “John is Mark’s child” (iv ) “Alice is John’s child” (v ) “The grandchild of a person is a child of a child of this person” 19
20
Chapter 2: Definite Logic Programs
These sentences may be formalized in two steps. First atomic formulas describing facts are introduced: child(tom, john) child(ann, tom) child(john, mark) child(alice, john)
(1) (2) (3) (4)
Applying this notation to the final sentence yields: “For all X and Y , grandchild(X, Y ) if there exists a Z such that child(X, Z) and child(Z, Y )”
(5)
This can be further formalized using quantifiers and the logical connectives “⊃” and “∧”, but to preserve the natural order of expression the implication is reversed and written “←”: ∀X ∀ Y (grandchild(X, Y ) ← ∃ Z (child(X, Z) ∧ child(Z, Y )))
(6)
This formula can be transformed into the following equivalent forms using the equivalences given in connection with Definition 1.15: ∀X ∀ Y ∀X ∀ Y ∀X ∀ Y ∀X ∀ Y
(grandchild(X, Y ) ∨ ¬ ∃ Z (child(X, Z) ∧ child(Z, Y ))) (grandchild(X, Y ) ∨ ∀ Z ¬ (child(X, Z) ∧ child(Z, Y ))) ∀ Z (grandchild(X, Y ) ∨ ¬ (child(X, Z) ∧ child(Z, Y ))) ∀ Z (grandchild(X, Y ) ← (child(X, Z) ∧ child(Z, Y )))
We now focus attention on the language of formulas exemplified by the example above. It consists of formulas of the form: A0 ← A1 ∧ · · · ∧ An
(where n ≥ 0)
or equivalently: A0 ∨ ¬A1 ∨ · · · ∨ ¬An where A0 , . . . , An are atomic formulas and all variables occurring in a formula are (implicitly) universally quantified over the whole formula. The formulas of this form are called definite clauses. Facts are definite clauses where n = 0. (Facts are sometimes called unit-clauses.) The atomic formula A0 is called the head of the clause whereas A1 ∧ · · · ∧ An is called its body. The initial example shows that definite clauses use a restricted form of existential quantification — the variables that occur only in body literals are existentially quantified over the body (though formally this is equivalent to universal quantification on the level of clauses).
2.2 Definite Programs and Goals
2.2
21
Definite Programs and Goals
The logic formulas derived above are special cases of a more general form, called clausal form. Definition 2.1 (Clause) A clause is a formula ∀(L1 ∨ · · · ∨ Ln ) where each Li is an atomic formula (a positive literal) or the negation of an atomic formula (a negative literal). As seen above, a definite clause is a clause that contains exactly one positive literal. That is, a formula of the form: ∀(A0 ∨ ¬A1 ∨ · · · ∨ ¬An ) The notational convention is to write such a definite clause thus: A0 ← A1 , . . . , An
(n ≥ 0)
If the body is empty (i.e. if n = 0) the implication arrow is usually omitted. Alternatively the empty body can be seen as a nullary connective which is true in every interpretation. (Symmetrically there is also a nullary connective 2 which is false in every interpretation.) The first kind of logic program to be discussed are programs consisting of a finite number of definite clauses: Definition 2.2 (Definite programs) A definite program is a finite set of definite clauses. To explain the use of logic formulas as programs, a general view of logic programming is presented in Figure 2.1. The programmer attempts to describe the intended model by means of declarative sentences (i.e. when writing a program he has in mind an algebraic structure, usually infinite, whose relations are to interpret the predicate symbols of the program). These sentences are definite clauses — facts and rules. The program is a set of logic formulas and it may have many models, including the intended model (Figure 2.1(a)). The concept of intended model makes it possible to discuss correctness of logic programs — a program P is incorrect iff the intended model is not a model of P . (Notice that in order to prove programs to be correct or to test programs it is necessary to have an alternative description of the intended model, independent of P .) The program will be used by the computer to draw conclusions about the intended model (Figure 2.1(b)). However, the only information available to the computer about the intended model is the program itself. So the conclusions drawn must be true in any model of the program to guarantee that they are true in the intended model (Figure 2.1(c)). In other words — the soundness of the system is a necessary condition. This will be discussed in Chapter 3. Before that, attention will be focused on the practical question of how a logic program is to be used. The set of logical consequences of a program is infinite. Therefore the user is expected to query the program selectively for various aspects of the intended model. There is an analogy with relational databases — facts explicitly describe elements of the relations while rules give intensional characterization of some other elements.
22
Chapter 2: Definite Logic Programs
model
intended model
model
P
(a)
P `F (b)
model
intended model
model
F
(c) Figure 2.1: General view of logic programming
2.2 Definite Programs and Goals
23
Since the rules may be recursive, the relation described may be infinite in contrast to the traditional relational databases. Another difference is the use of variables and compound terms. This chapter considers only “queries” of the form: ∀(¬(A1 ∧ · · · ∧ Am )) Such formulas are called definite goals and are usually written as: ← A1 , . . . , Am where Ai ’s are atomic formulas called subgoals. The goal where m = 0 is denoted 21 and called the empty goal. The logical meaning of a goal can be explained by referring to the equivalent universally quantified formula: ∀X1 · · · ∀Xn ¬(A1 ∧ · · · ∧ Am ) where X1 , . . . , Xn are all variables that occur in the goal. This is equivalent to: ¬ ∃X1 · · · ∃Xn (A1 ∧ · · · ∧ Am ) This, in turn, can be seen as an existential question and the system attempts to deny it by constructing a counter-example. That is, it attempts to find terms t1 , . . . , tn such that the formula obtained from A1 ∧ · · · ∧ Am when replacing the variable Xi by ti (1 ≤ i ≤ n), is true in any model of the program, i.e. to construct a logical consequence of the program which is an instance of a conjunction of all subgoals in the goal. By giving a definite goal the user selects the set of conclusions to be constructed. This set may be finite or infinite. The problem of how the machine constructs it will be discussed in Chapter 3. The section is concluded with some examples of queries and the answers obtained to the corresponding goals in a typical Prolog system. Example 2.3 Referring to the family-example in Section 2.1 the user may ask the following queries (with the corresponding goal): Query “Is Ann a child of Tom?” “Who is a grandchild of Ann?” “Whose grandchild is Tom?” “Who is a grandchild of whom?”
Goal ← child(ann, tom) ← grandchild(X, ann) ← grandchild(tom, X) ← grandchild(X, Y )
The following answers are obtained: • Since there are no variables in the first goal the answer is simply “yes”; • Since the program contains no information about grandchildren of Ann the answer to the second goal is “no one” (although most Prolog implementations would answer simply “no”; 1 Of course, formally it is not correct to write ← A , . . . , A m since “←” should have a formula 1 also on the left-hand side. The problem becomes even more evident when m = 0 because then the right-hand side disappears as well. However, formally the problem can be viewed as follows — a definite goal has the form ∀(¬(A1 ∧ · · · ∧ Am )) which is equivalent to ∀(2 ∨ ¬(A1 ∧ · · · ∧ Am ∧ )). A nonempty goal can thus be viewed as the formula ∀(2 ← (A1 ∧ · · · ∧ Am )). The empty goal can be viewed as the formula 2 ← which is equivalent to 2.
24
Chapter 2: Definite Logic Programs • Since Tom is the grandchild of Mark the answer is X = mark in reply to the third goal; • The final goal yields three answers: X = tom X = alice X = ann
Y = mark Y = mark Y = john
It is also possible to ask more complicated queries, for example “Is there a person whose grandchildren are Tom and Alice?”, expressed formally as: ← grandchild(tom, X), grandchild(alice, X) whose (expected) answer is X = mark.
2.3
The Least Herbrand Model
Definite programs can only express positive knowledge — both facts and rules say which elements of a structure are in a relation, but they do not say when the relations do not hold. Therefore, using the language of definite programs, it is not possible to construct contradictory descriptions, i.e. unsatisfiable sets of formulas. In other words, every definite program has a model. This section discusses this matter in more detail. It shows also that every definite program has a well defined least model. Intuitively this model reflects all information expressed by the program and nothing more. We first focus attention on models of a special kind, called Herbrand models. The idea is to abstract from the actual meanings of the functors (here, constants are treated as 0-ary functors) of the language. More precisely, attention is restricted to the interpretations where the domain is the set of variable-free terms and the meaning of every ground term is the term itself. After all, it is a common practice in databases — the constants tom and ann may represent persons but the database describes relations between the persons by handling relations between the terms (symbols) no matter whom they represent. The formal definition of such domains follows and is illustrated by two simple examples. Definition 2.4 (Herbrand universe, Herbrand base) Let A be an alphabet containing at least one constant symbol. The set UA of all ground terms constructed from functors and constants in A is called the Herbrand universe of A. The set BA of all ground, atomic formulas over A is called the Herbrand base of A. The Herbrand universe and Herbrand base are often defined for a given program. In this case it is assumed that the alphabet of the program consists of exactly those symbols which appear in the program. It is also assumed that the program contains at least one constant (since otherwise, the domain would be empty). Example 2.5 Consider the following definite program P :
2.3 The Least Herbrand Model
25
odd(s(0)). odd(s(s(X))) ← odd(X). The program contains one constant (0) and one unary functor (s). Consequently the Herbrand universe looks as follows: UP = {0, s(0), s(s(0)), s(s(s(0))), . . .} Since the program contains only one (unary) predicate symbol (odd) it has the following Herbrand base: BP = {odd(0), odd(s(0)), odd(s(s(0))), . . .}
Example 2.6 Consider the following definite program P : owns(owner(corvette ), corvette). happy(X) ← owns(X, corvette). In this case the Herbrand universe UP consists of the set: {corvette, owner(corvette), owner(owner(corvette )), . . .} and the Herbrand base BP of the set: {owns(s, t) | s, t ∈ UP } ∪ {happy(s) | s ∈ UP }
Definition 2.7 (Herbrand interpretations) A Herbrand interpretation of P is an interpretation = such that: • the domain of = is UP ; • for every constant c, c= is defined to be c itself; • for every n-ary functor f the function f= is defined as follows f= (x1 , . . . , xn ) := f (x1 , . . . , xn ) That is, the function f= applied to n ground terms composes them into the ground term with the principal functor f ; • for every n-ary predicate symbol p the relation p= is a subset of UPn (the set of all n-tuples of ground terms). Thus Herbrand interpretations have predefined meanings of functors and constants and in order to specify a Herbrand interpretation it suffices to list the relations associated with the predicate symbol. Hence, for an n-ary predicate symbol p and a Herbrand interpretation = the meaning p= of p consists of the following set of ntuples: {ht1 , . . . , tn i ∈ UPn | = |= p(t1 , . . . , tn )}.
26
Chapter 2: Definite Logic Programs
Example 2.8 One possible interpretation of the program P in Example 2.5 is odd= = {hs(0)i, hs(s(s(0)))i}. A Herbrand interpretation can be specified by giving a family of such relations (one for every predicate symbol). Since the domain of a Herbrand interpretation is the Herbrand universe the relations are sets of tuples of ground terms. One can define all of them at once by specifying a set of labelled tuples, where the labels are predicate symbols. In other words: A Herbrand interpretation = can be seen as a subset of the Herbrand base (or a possibly infinite relational database), namely {A ∈ BP | = |= A}. Example 2.9 Consider some alternative Herbrand interpretations for P of Example 2.5. =1 =2 =3 =4 =5
:= := := := = :=
? {odd(s(0))} {odd(s(0)), odd(s(s(0)))} {odd(sn (0)) | n ∈ {1, 3, 5, 7, . . .}} {odd(s(0)), odd(s(s(s(0)))), . . .} BP
Definition 2.10 (Herbrand model) A Herbrand model of a set of (closed) formulas is a Herbrand interpretation which is a model of every formula in the set. It turns out that Herbrand interpretations and Herbrand models have two attractive properties. The first is pragmatic: In order to determine if a Herbrand interpretation = is a model of a universally quantified formula ∀F it suffices to check if all ground instances of F are true in =. For instance, to check if A0 ← A1 , . . . , An is true in = it suffices to show that if (A0 ← A1 , . . . , An )θ is a ground instance of A0 ← A1 , . . . , An and A1 θ, . . . , An θ ∈ = then A0 θ ∈ =. Example 2.11 Clearly =1 cannot be a model of P in Example 2.5 as it is not a Herbrand model of odd(s(0)). However, =2 , =3 , =4 , =5 are all models of odd(s(0)) since odd(s(0)) ∈ =i , (2 ≤ i ≤ 5). Now, =2 is not a model of odd(s(s(X))) ← odd(X) since there is a ground instance of the rule — namely odd(s(s(s(0)))) ← odd(s(0)) — such that all premises are true: odd(s(0)) ∈ =2 , but the conclusion is false: odd(s(s(s(0)))) 6∈ =2 . By a similar reasoning it follows that =3 is not a model of the rule. However, =4 is a model also of the rule; let odd(s(s(t))) ← odd(t) be any ground instance of the rule where t ∈ UP . Clearly, odd(s(s(t))) ← odd(t) is true if odd(t) 6∈ =4 (check with Definition 1.6). Furthermore, if odd(t) ∈ =4 then it must also hold that odd(s(s(t))) ∈ =4 (cf. the the definition of =4 above) and hence odd(s(s(t))) ← odd(t) is true in =4 . Similar reasoning proves that =5 is also a model of the program. The second reason for focusing on Herbrand interpretations is more theoretical. For the restricted language of definite programs, it turns out that in order to determine whether an atomic formula A is a logical consequence of a definite program P it suffices to check that every Herbrand model of P is also a Herbrand model of A.
2.3 The Least Herbrand Model
27
Theorem 2.12 Let P be a definite program and G a definite goal. If =0 is a model of P ∪ {G} then = := {A ∈ BP | =0 |= A} is a Herbrand model of P ∪ {G}. Proof : Clearly, = is a Herbrand interpretation. Now assume that =0 is a model and that = is not a model of P ∪ {G}. In other words, there exists a ground instance of a clause or a goal in P ∪ {G}: A0 ← A1 , . . . , Am
(m ≥ 0)
which is not true in = (A0 = 2 in case of a goal). Since this clause is false in = then A1 , . . . , Am are all true and A0 is false in =. Hence, by the definition of = we conclude that A1 , . . . , Am are true and A0 is false in =0 . This contradicts the assumption that =0 is a model. Hence = is a model of P ∪ {G}. Notice that the form of P in Theorem 2.12 is restricted to definite programs. In the general case, nonexistence of a Herbrand model of a set of formulas P does not mean that P is unsatisfiable. That is, there are sets of formulas P which do not have a Herbrand model but which have other models.2 Example 2.13 Consider the formulas {¬p(a), ∃ X p(X)} where UP := {a} and BP := {p(a)}. Clearly, there are only two Herbrand interpretations — the empty set and BP itself. The former is not a model of the second formula. The latter is a model of the second formula but not of the first. However, it is not very hard to find a model of the formulas — let the domain be the natural numbers, assign 0 to the constant a and the relation {h1i, h3i, h5i, . . .} to the predicate symbol p (i.e. let p denote the “odd”-relation). Clearly this is a model since “0 is not odd” and “there exists a natural number which is odd, e.g. 1”. Notice that the Herbrand base of a definite program P always is a Herbrand model of the program. To check that this is so, simply take an arbitrary ground instance of any clause A0 ← A1 , . . . , Am in P . Clearly, all A0 , . . . , Am are in the Herbrand base. Hence the formula is true. However, this model is rather uninteresting — every n-ary predicate of the program is interpreted as the full n-ary relation over the domain of ground terms. More important is of course the question — what are the interesting models of the program? Intuitively there is no reason to expect that the model includes more ground atoms than those which follow from the program. By the analogy to databases — if John is not in the telephone directory he probably has no telephone. However, the directory gives only positive facts and if John has a telephone it is not a contradiction to what is said in the directory. The rest of this section is organized as follows. First it is shown that there exists a unique minimal model called the least Herbrand model of a definite program. Then it is shown that this model really contains all positive information present in the program. The Herbrand models of a definite program are subsets of its Herbrand base. Thus the set-inclusion is a natural ordering of such models. In order to show the existence of least models with respect to set-inclusion it suffices to show that the intersection of all Herbrand models is also a (Herbrand) model. 2 More
generally the result of Theorem 2.12 would hold for any set of clauses.
28
Chapter 2: Definite Logic Programs
Theorem 2.14 (Model intersection property) Let M be a non-empty T family of Herbrand models of a definite program P . Then the intersection = := M is a Herbrand model of P . Proof : Assume that = is not a model of P . Then there exists a ground instance of a clause of P : A0 ← A1 , . . . , Am
(m ≥ 0)
which is not true in =. This implies that = contains A1 , . . . , Am but not A0 . Then A1 , . . . , Am are elements of every interpretation of the family M . Moreover there must be at least one model =i ∈ M such that A0 6∈ =i . Thus A0 ← A1 , . . . , Am is not true in this =i . Hence =i is not a model of the program, which contradicts the assumption. This concludes the proof that the intersection of any set of Herbrand models of a program is also a Herbrand model. Thus by taking the intersection of all Herbrand models (it is known that every definite program P has at least one Herbrand model — namely BP ) the least Herbrand model of the definite program is obtained. Example 2.15 Let P be the definite program {male(adam), female(eve)} with obvious intended interpretation. P has the following four Herbrand models: {male(adam), female(eve)} {male(adam), male(eve), female(eve)} {male(adam), female(eve), female(adam)} {male(adam), male(eve), female(eve), female(adam)} It is not very hard to see that any intersection of these yields a Herbrand model. However, all but the first model contain atoms incompatible with the intended one. Notice also that the intersection of all four models yields a model which corresponds to the intended model. This example indicates a connection between the least Herbrand model and the intended model of a definite program. The intended model is an abstraction of the world to be described by the program. The world may be richer than the least Herbrand model. For instance, there may be more female individuals than just Eve. However, the information not included explicitly (via facts) or implicitly (via rules) in the program cannot be obtained as an answer to a goal. The answers correspond to logical consequences of the program. Ideally, a ground atomic formula p(t1 , . . . , tn ) is a logical consequence of the program iff, in the intended interpretation =, ti denotes the individual xi and hx1 , . . . , xn i ∈ p= . The set of all such ground atoms can be seen as a “coded” version of the intended model. The following theorem relates this set to the least Herbrand model. Theorem 2.16 The least Herbrand model MP of a definite program P is the set of all ground atomic logical consequences of the program. That is, MP = {A ∈ BP | P |= A}.
2.4 Construction of Least Herbrand Models
29
Proof : Show first MP ⊇ {A ∈ BP | P |= A}: It is easy to see that every ground atom A which is a logical consequence of P is an element of MP . Indeed, by the definition of logical consequence A must be true in MP . On the other hand, the definition of Herbrand interpretation states that A is true in MP iff A is an element of MP . Then show that MP ⊆ {A ∈ BP | P |= A}: Assume that A is in MP . Hence it is true in every Herbrand model of P . Assume that it is not true in some non-Herbrand model =0 of P . But we know (see Theorem 2.12) that the set = of all ground atomic formulas which are true in =0 is a Herbrand model of P . Hence A cannot be an element of =. This contradicts the assumption that there exists a model of P where A is false. Hence A is true in every model of P , that is P |= A, which concludes the proof. The model intersection property expressed by Theorem 2.14 does not hold for arbitrary formulas as illustrated by the following example. Example 2.17 Consider the formula p(a) ∨ q(b). Clearly, both {p(a)} and {q(b)} are Herbrand models of the formula. However, the intersection {p(a)} ∩ {q(b)} = ? is not a model. The two models are examples of minimal models — that is, one cannot remove any element from the model and still have a model. However, there is no least model — that is, a unique minimal model.
2.4
Construction of Least Herbrand Models
The question arises how the least Herbrand model can be constructed, or approximated by successive enumeration of its elements. The answer to this question is given by a fixed point approach to the semantics of definite programs. (A fixpoint of a function f : D → D is an element x ∈ D such that f (x) = x.) This section gives only a sketch of the construction. The discussion of the relevant theory is outside of the scope of this book. However, the intuition behind the construction is the following: A definite program consists of facts and rules. Clearly, all ground instances of the facts must be included in every Herbrand model. If a Herbrand interpretation = does not include a ground instance of a fact A of the program then A is not true in = and = is not a model. Next, consider a rule A0 ← A1 , . . . , Am where (m > 0). This rule states that whenever A1 , . . . , Am are true then so is A0 . In other words, take any ground instance (A0 ← A1 , . . . , Am )θ of the rule. If = includes A1 θ, . . . , Am θ it must also include A0 θ in order to be a model. Consider the set =1 of all ground instances of facts in the program. It is now possible to use every instance of each rule to augment =1 with new elements which necessarily must belong to every model. In that way a new set =2 is obtained which can be used again to generate more elements which must belong to the model. This process is repeated as long as new elements are generated. The new elements added to =i+1 are those which must follow immediately from =i . The construction outlined above can be formally defined as an iteration of a transformation TP on Herbrand interpretations of the program P . The operation is called the immediate consequence operator and is defined as follows: Definition 2.18 (Immediate consequence operator) Let ground(P ) be the set of all ground instances of clauses in P . TP is a function on Herbrand interpretations
30
Chapter 2: Definite Logic Programs
of P defined as follows: TP (I) := {A0 | A0 ← A1 , . . . , Am ∈ ground(P ) ∧ {A1 , . . . , Am } ⊆ I}
For definite programs it can be shown that there exists a least interpretation = such that TP (=) = = and that = is identical with the least Herbrand model MP . Moreover, MP is the limit of the increasing, possibly infinite sequence of iterations:
?, TP (?), TP (TP (?)), TP (TP (TP (?))), . . . There is a standard notation used to denote elements of the sequence of interpretations constructed for P . Namely: TP ↑ 0 := ? TP ↑ (i + 1) := TP (TP ↑ i) ∞ [ TP ↑ ω := TP ↑ i i=0
The following example illustrates the construction: Example 2.19 Consider again the program of Example 2.5. TP ↑ 0 = ? TP ↑ 1 = {odd(s(0))} TP ↑ 2 = {odd(s(s(s(0)))), odd(s(0))} .. . TP ↑ ω = {odd(sn (0)) | n ∈ {1, 3, 5, . . .}}
As already mentioned above it has been established that the set constructed in this way is identical to the least Herbrand model. Theorem 2.20 Let P be a definite program and MP its least Herbrand model. Then: • MP is the least Herbrand interpretation such that TP (MP ) = MP (i.e. it is the least fixpoint of TP ). • MP = TP ↑ ω.
For additional details and proofs see for example Apt (1990), Lloyd (1987) or van Emden and Kowalski (1976).
Exercises
31
Exercises 2.1 Rewrite the following formulas in the form A0 ← A1 , . . . , Am : ∀X(p(X) ∨ ¬q(X)) ∀X(p(X) ∨ ¬∃ Y (q(X, Y ) ∧ r(X))) ∀X(¬p(X) ∨ (q(X) ⊃ r(X))) ∀X(r(X) ⊃ (q(X) ⊃ p(X))) 2.2 Formalize the following scenario as a definite program: Basil owns Fawlty Towers. Basil and Sybil are married. Polly and Manuel are employees at Fawlty Towers. Smith and Jones are guests at Fawlty Towers. All hotel-owners and their spouses serve all guests at the hotel. All employees at a hotel serve all guests at the hotel. All employees dislike the owner of the workplace. Basil dislikes Manuel. Then ask the queries “Who serves who?” and “Who dislikes who?”. 2.3 Give the Herbrand universe and Herbrand base of the following definite program: p(f (X)) ← q(X, g(X)). q(a, g(b)). q(b, g(b)). 2.4 Give the Herbrand universe and Herbrand base of the following definite program: p(s(X), Y, s(Z)) ← p(X, Y, Z). p(0, X, X). 2.5 Consider the Herbrand universe consisting of the constants a, b, c and d. Let = be the Herbrand interpretation: {p(a), p(b), q(a), q(b), q(c), q(d)} Which of the following formulas are true in =? (1) (2) (3) (4) (5)
∀Xp(X) ∀Xq(X) ∃X(q(X) ∧ p(X)) ∀X(q(X) ⊃ p(X)) ∀X(p(X) ⊃ q(X))
2.6 Give the least Herbrand model of the program in exercise 2.3. 2.7 Give the least Herbrand model of the program in exercise 2.4. Hint: the model is infinite, but a certain pattern can be spotted when using the TP -operator.
32
Chapter 2: Definite Logic Programs 2.8 Consider the following program: p(0). p(s(X)) ← p(X). Show that p(sn (0)) ∈ TP ↑ m iff n < m. 2.9 Let P be a definite program and = a Herbrand interpretation. Show that = is a model of P iff TP (=) ⊆ =.
Chapter 3 SLD-Resolution
This chapter introduces the inference mechanism which is the basis of most logic programming systems. The idea is a special case of the inference rule called the resolution principle — an idea that was first introduced by J. A. Robinson in the mid-sixties for a richer language than definite programs. As a consequence, only a specialization of this rule, that applies to definite programs, is presented here. For reasons to be explained later, it will be called the SLD-resolution principle. In the previous chapter the model-theoretic semantics of definite programs was discussed. The SLD-resolution principle makes it possible to draw correct conclusions from the program, thus providing a foundation for a logically sound operational semantics of definite programs. This chapter first defines the notion of SLD-resolution and then shows its correctness with respect to the model-theoretic semantics. Finally SLD-resolution is shown to be an instance of a more general notion involving the construction of proof trees.
3.1
Informal Introduction
Every inference rule of a logical system formalizes some natural way of reasoning. The presentation of the SLD-resolution principle is therefore preceded by an informal discussion about the underlying reasoning techniques. The sentences of logic programs have a general structure of logical implication: A0 ← A1 , . . . , An
(n ≥ 0)
where A0 , . . . , An are atomic formulas and where A0 may be absent (in which case it is a goal clause). Consider the following definite program that describes a world where “parents of newborn children are proud”, “Adam is the father of Mary” and “Mary is newborn”: 33
34
Chapter 3: SLD-Resolution proud (X) ← parent(X, Y ), newborn(Y ). parent(X, Y ) ← father (X, Y ). parent(X, Y ) ← mother (X, Y ). father (adam, mary). newborn(mary).
Notice that this program describes only “positive knowledge” — it does not state who is not proud. Nor does it convey what it means for someone not to be a parent. The problem of expressing negative knowledge will be investigated in detail in Chapter 4 when extending definite programs with negation. Say now that we want to ask the question “Who is proud?”. The question concerns the world described by the program P , that is, the intended model of P . We would of course like to see the answer “Adam” to this question. However, as discussed in the previous chapters predicate logic does not provide the means for expressing this type of interrogative sentences; only declarative ones. Therefore the question may be formalized as the goal clause: ← proud (Z)
(G0 )
which is an abbreviation for ∀ Z ¬proud (Z) which in turn is equivalent to: ¬ ∃ Z proud (Z) whose reading is “Nobody is proud”. That is, a negative answer to the query above. The aim now is to show that this answer is a false statement in every model of P (and in particular in the intended model). Then by Proposition 1.13 it can be concluded that P |= ∃ Z proud (Z). Alas this would result only in a “yes”-answer to the original question, while the expected answer is “Adam”. Thus, the objective is rather to find a substitution θ such that the set P ∪ {¬ proud (Z)θ} is unsatisfiable, or equivalently such that P |= proud (Z)θ. The starting point of reasoning is the assumption G0 — “For any Z, Z is not proud”. Inspection of the program reveals a rule describing one condition for someone to be proud: proud (X) ← parent(X, Y ), newborn(Y ).
(C0 )
Its equivalent logical reading is: ∀(¬proud (X) ⊃ ¬(parent (X, Y ) ∧ newborn(Y ))) Renaming X into Z, elimination of universal quantification and the use of modus ponens with respect to G0 yields: ¬ (parent (Z, Y ) ∧ newborn(Y )) or equivalently: ← parent (Z, Y ), newborn(Y ).
(G1 )
3.1 Informal Introduction
35
Thus, one step of reasoning amounts to replacing a goal G0 by another goal G1 which is true in any model of P ∪ {G0 }. It now remains to be shown that P ∪ {G1 } is unsatisfiable. Note that G1 is equivalent to: ∀ Z ∀ Y (¬parent (Z, Y ) ∨ ¬newborn(Y )) Thus G1 can be shown to be unsatisfiable with P if in every model of P there is some individual who is a parent of a newborn child. Thus, check first whether there are any parents at all. The program contains a clause: parent (X, Y ) ← father (X, Y ).
(C1 )
which is equivalent to: ∀(¬parent (X, Y ) ⊃ ¬father (X, Y )) Thus, G1 reduces to: ← father (Z, Y ), newborn(Y ).
(G2 )
The new goal G2 can be shown to be unsatisfiable with P if in every model of P there is some individual who is a father of a newborn child. The program states that “Adam is the father of Mary”: father (adam, mary).
(C2 )
Thus it remains to be shown that “Mary is not newborn” is unsatisfiable together with P: ← newborn(mary).
(G3 )
But the program also contains a fact: newborn(mary).
(C3 )
equivalent to ¬newborn(mary) ⊃ 2 leading to a refutation:
2
(G4 )
The way of reasoning used in this example is as follows: to show existence of something, assume the contrary and use modus ponens and elimination of the universal quantifier to find a counter-example for the assumption. This is a general idea to be used in computations of logic programs. As illustrated above, a single computation (reasoning) step transforms a set of atomic formulas — that is, a definite goal — into a new set of atoms. (See Figure 3.1.) It uses a selected atomic formula p(s1 , . . . , sn ) of the goal and a selected program clause of the form p(t1 , . . . , tn ) ← A1 , . . . , Am (where m ≥ 0 and A1 , . . . , Am are atoms) to find a common instance of p(s1 , . . . , sn ) and p(t1 , . . . , tn ). In other words a substitution θ is constructed such that p(s1 , . . . , sn )θ and p(t1 , . . . , tn )θ are identical. Such a substitution is called a unifier and the problem of finding unifiers will be discussed in the next section. The new goal is constructed from the old one by replacing the selected atom by the set of body atoms of the clause and applying θ to all
36
Chapter 3: SLD-Resolution
← proud (Z).
proud (X) ← parent (X, Y ), newborn(Y ). ← parent (Z, Y ), newborn(Y ). parent (X, Y ) ← father (X, Y ). ← father (Z, Y ), newborn(Y ). father (adam, mary). ← newborn(mary). newborn(mary). 2 Figure 3.1: Refutation of ← proud (Z).
atoms obtained in that way. This basic computation step can be seen as an inference rule since it transforms logic formulas. It will be called the resolution principle for definite programs or SLD-resolution principle. As illustrated above it combines in a special way modus ponens with the elimination rule for the universal quantifier. At the last step of reasoning the empty goal, corresponding to falsity, is obtained. The final conclusion then is the negation of the initial goal. Since this goal is of the form ∀ ¬(A1 ∧ · · · ∧ Am ), the conclusion is equivalent (by DeMorgan’s laws) to the formula ∃(A1 ∧ · · · ∧ Am ). The final conclusion can be obtained by the inference rule known as reductio ad absurdum. Every step of reasoning produces a substitution. Unsatisfiability of the original goal ← A1 , . . . , Am with P is demonstrated in k steps by showing that its instance: ← (A1 , . . . , Am )θ1 · · · θk is unsatisfiable, or equivalently that: P |= (A1 ∧ · · · ∧ Am )θ1 · · · θk In the example discussed, the goal “Nobody is proud” is unsatisfiable with P since its instance “Adam is not proud” is unsatisfiable with P . In other words — in every model of P the sentence “Adam is proud” is true. It is worth noticing that the unifiers may leave some variables unbound. In this case the universal closure of (A1 ∧ · · · ∧ Am )θ1 · · · θk is a logical consequence of P . Examples of such answers will appear below. Notice also that generally the computation steps are not deterministic — any atom of a goal may be selected and there may be several clauses matching the selected atom. Another potential source of non-determinism concerns the existence of alternative unifiers for two atoms. These remarks suggest that it may be possible to construct (sometimes infinitely) many solutions, i.e. counter-examples for the initial goal. On the other hand it may also happen that the selected atom has no matching clause.
3.2 Unification
37
If so, it means that, using this method, it is not possible to construct any counterexample for the initial goal. The computation may also loop without producing any solution.
3.2
Unification
As demonstrated in the previous section, one of the main ingredients in the inference mechanism is the process of making two atomic formulas syntactically equivalent. Before defining the notion of SLD-resolution we focus on this process, called unification, and give an algorithmic solution — a procedure that takes two atomic formulas as input, and either shows how they can be instantiated to identical atoms or, reports a failure. Before considering the problem of unifying atoms (and terms), consider an ordinary equation over the natural numbers (N ) such as: . 2x + 3 = 4y + 7
(5)
The equation has a set of solutions; that is, valuations ϕ: {x, y} → N such that ϕ= (2x+ 3) = ϕ= (4y + 7) where = is the standard interpretation of the arithmetic symbols. In this particular example there are infinitely many solutions ({x 7→ 2, y 7→ 0} and {x 7→ 4, y 7→ 1} etc.) but by a sequence of syntactic transformations that preserve the set of all solutions the equation may be transformed into an new equation that compactly represents all solutions to the original equation: . x = 2(y + 1)
(6)
The transformations exploit domain knowledge (such as commutativity, associativity etc.) specific to the particular interpretation. In a logic program there is generally no such knowledge available and the question arises how to compute the solutions of an equation without any knowledge about the interpretation of the symbols. For example: . f (X, g(Y )) = f (a, g(X))
(7)
Clearly it is no longer possible to apply all the transformations that were applied above since the interpretation of f /2, g/1 is no longer fixed. However, any solution of the equations: . . {X = a, g(Y ) = g(X)}
(8)
must clearly be a solution of equation (7). Similarly, any solution of: . . {X = a, Y = X}
(9)
must be a solution of equations (8). Finally any solution of: . . {X = a, Y = a}
(10)
is a solution of (9). By analogy to (6) this is a compact representation of some solutions to equation (7). However, whether it represents all solution depends on how the
38
Chapter 3: SLD-Resolution
symbols f /2, g/1 and a are interpreted. For example, if f /2 denotes integer addition, g/1 the successor function and a the integer zero, then (10) represents only one solution to equation (7). However, equation (7) has infinitely many integer solutions — any ϕ such that ϕ(Y ) = 0 is a solution. On the other hand, consider a Herbrand interpretation =; Solving of an equation . s = t amounts to finding a valuation ϕ such that ϕ= (s) = ϕ= (t). Now a valuation in the Herbrand domain is a mapping from variables of the equations to ground terms (that is, a substitution) and the interpretation of a ground term is the term itself. Thus, a solution in the Herbrand domain is a grounding substitution ϕ such that sϕ and tϕ are identical ground terms. This brings us to the fundamental concept of unification and unifiers: Definition 3.1 (Unifier) Let s and t be terms. A substitution θ such that sθ and tθ are identical (denoted sθ = tθ) is called a unifier of s and t. The search for a unifier of two terms, s and t, will be viewed as the process of solving . . . the equation s = t. Therefore, more generally, if {s1 = t1 , . . . , sn = tn } is a set of equations, then θ is called a unifier of the set if si θ = ti θ for all 1 ≤ i ≤ n. For instance, the substitution {X/a, Y /a} is a unifier of equation (7). It is also a unifier of (8)–(10). In fact, it is the only unifier as long as “irrelevant” variables are not introduced. (For instance, {X/a, Y /a, Z/a} is also a unifier.) The transformations informally used in steps (7)–(10) preserve the set of all solutions in the Herbrand domain. (The full set of transformations will soon be presented.) Note that a solution to a set of equations is a (grounding) unifier. Thus, if a set of equations has a unifier then the set also has a solution. However, not all sets of equations have a solution/unifier. For instance, the set . {sum(1, 1) = 2} is not unifiable. Intuitively sum may be thought of as integer addition, but bear in mind that the symbols have no predefined interpretation in a logic program. (In Chapters 13–14 more powerful notions of unification are discussed.) It is often the case that a set of equations have more than one unifier. For in. stance, both {X/g(Z), Y /Z} and {X/g(a), Y /a, Z/a} are unifiers of the set {f (X, Y ) = f (g(Z), Z)}. Under the first unifier the terms instantiate to f (g(Z), Z) and under the second unifier the terms instantiate to f (g(a), a). The second unifier is in a sense more restrictive than the first, as it makes the two terms ground whereas the first still provides room for some alternatives in that is does not specify how Z should be bound. We say that {X/g(Z), Y /Z} is more general than {X/g(a), Y /a, Z/a}. More formally this can be expressed as follows: Definition 3.2 (Generality of substitutions) A substitution θ is said to be more general than a substitution σ (denoted σ θ) iff there exists a substitution ω such that σ = θω. Definition 3.3 (Most general unifier) A unifier θ is said to be a most general unifier (mgu) of two terms iff θ is more general than any other unifier of the terms. . . Definition 3.4 (Solved form) A set of equations {X1 = t1 , . . . , Xn = tn } is said to be in solved form iff X1 , . . . , Xn are distinct variables none of which appear in t1 , . . . , tn .
3.2 Unification
39
There is a close correspondence between a set of equations in solved form and the most general unifier(s) of that set as shown by the following theorem: . . Proposition 3.5 Let {X1 = t1 , . . . , Xn = tn } be a set of equations in solved form. Then {X1 /t1 , . . . , Xn /tn } is an (idempotent) mgu of the solved form. Proof : First define: . . E := {X1 = t1 , . . . , Xn = tn } θ := {X1 /t1 , . . . , Xn /tn } Clearly θ is an idempotent unifier of E. It remains to be shown that θ is more general than any other unifier of E. Thus, assume that σ is a unifier of E. Then Xi σ = ti σ for 1 ≤ i ≤ n. It must follow that Xi /ti σ ∈ σ for 1 ≤ i ≤ n. In addition σ may contain some additonal pairs Y1 /s1 , . . . , Ym /sm such that {X1 , . . . , Xn } ∩ {Y1 , . . . , Ym } = ?. Thus, σ is of the form: {X1 /t1 σ, . . . , Xn /tn σ, Y1 /s1 , . . . , Ym /sm } Now θσ = σ. Thus, there exists a substitution ω (viz. σ) such that σ = θω. Therefore, θ is an idempotent mgu. Definition 3.6 (Equivalence of sets of equations) Two sets of equations E1 and E2 are said to be equivalent if they have the same set of unifiers. Note that two equivalent sets of equations must have the same set of solutions in any Herbrand interpretation. The definition can be used as follows: to compute a most general unifier mgu(s, t) . of two terms s and t, first try to transform the equation {s = t} into an equivalent solved form. If this fails then mgu(s, t) = failure. However, if there is a solved form . . {X1 = t1 , . . . , Xn = tn } then mgu(s, t) = {X1 /t1 , . . . , Xn /tn }. Figure 3.2 presents a (non-deterministic) algorithm which takes as input a set of equations E and terminates returning either a solved form equivalent to E or failure if no such solved form exists. Note that constants are viewed as function symbols of . arity 0. Thus, if an equation c = c gets selected, the equation is simply removed by case 1. Before proving the correctness of the algorithm some examples are used to illustrate the idea: . Example 3.7 The set {f (X, g(Y )) = f (g(Z), Z)} has a solved form since: . . . {f (X, g(Y )) = f (g(Z), Z)} ⇒ {X = g(Z), g(Y ) = Z} . . ⇒ {X = g(Z), Z = g(Y )} . . ⇒ {X = g(g(Y )), Z = g(Y )} . The set {f (X, g(X), b) = f (a, g(Z), Z)}, on the other hand, does not have a solved form since: . . . . {f (X, g(X), b) = f (a, g(Z), Z)} ⇒ {X = a, g(X) = g(Z), b = Z} . . . ⇒ {X = a, g(a) = g(Z), b = Z}
40
Chapter 3: SLD-Resolution
Input: A set E of equations. Output: An equivalent set of equations in solved form or failure. repeat . select an arbitrary s = t ∈ E; . case s = t of . f (s1 , . . . , sn ) = f (t1 , . . . , tn ) where n ≥ 0 ⇒ . . replace equation by s1 = t1 , . . . , sn = tn ; % case . f (s1 , . . . , sm ) = g(t1 , . . . , tn ) where f /m 6= g/n ⇒ halt with failure; % case . X =X ⇒ remove the equation; % case . t = X where t is not a variable ⇒ . replace equation by X = t; % case . X = t where X 6= t and X has more than one occurrence in E if X is a proper subterm of t then halt with failure % case else replace all other occurrences of X by t; % case esac until no action is possible on any equation in E; halt with E;
1 2 3 4 ⇒ 5a 5b
Figure 3.2: Solved form algorithm
⇒ ⇒ ⇒ ⇒
. . . {X = a, a = Z, b = Z} . . . {X = a, Z = a, b = Z} . . . {X = a, Z = a, b = a} failure
. The algorithm fails since case 2 applies to b = a. Finally consider: . {f (X, g(X)) = f (Z, Z)} ⇒ ⇒ ⇒ ⇒
. . {X = Z, g(X) = Z} . . {X = Z, g(Z) = Z} . . {X = Z, Z = g(Z)} failure
The set does not have a solved form since Z is a proper subterm of g(Z). Theorem 3.8 The solved form algorithm in Figure 3.2 terminates and returns an equivalent solved form or failure if no such solved form exists. Proof : First consider termination: Note that case 5b is the only case that may increase the number of symbol occurrences in the set of equations. However, case 5b can be applied at most once for each variable X. Thus, case 5b can be applied only a finite
3.2 Unification
41
number of times and may introduce only a finite number of new symbol occurrences. Case 2 and case 5a terminate immediately and case 1 and 3 strictly decrease the number of symbol occurrences in the set. Since case 4 cannot be applied indefinitely, but has to be intertwined with the other cases it follows that the algorithm always terminates. It should be evident that the algorithm either returns failure or a set of equations in solved form. Thus, it remains to be shown that each iteration of the algorithm preserves equivalence between successive sets of equations. It is easy to see that if case 2 or 5a apply to some equation in: . . {s1 = t1 , . . . , sn = tn }
(E1 )
then the set cannot possibly have a unifier. It is also easy to see that if any of case 1, 3 or 4 apply, then the new set of equations has the same set of unifiers. Finally assume . that case 5b applies to some equation si = ti . Then the new set is of the form: . . . . . {s1 θ = t1 θ, . . . , si−1 θ = ti−1 θ, si = ti , si+1 θ = ti+1 θ, . . . sn θ = tn θ}
(E2 )
where θ := {si /ti }. First assume that σ is a unifier of E1 — that is, sj σ = tj σ for every 1 ≤ j ≤ n. In particular, it must hold that si σ = ti σ. Since si is a variable which is not a subterm of ti it must follow that si /ti σ ∈ σ. Moreover, θσ = σ and it therefore follows that σ is a unifier also of E2 . Next, assume that σ is a unifier of E2 . Thus, si /ti σ ∈ σ and θσ = σ which must then be a unifier also of E1 . The algorithm presented in Figure 3.2 may be very inefficient. One of the reasons is case 5a; That is, checking if a variable X occurs inside another term t. This is often referred to as the occur-check. Assume that the time of occur-check is linear with respect to the size |t| of t.1 Consider application of the solved form algorithm to the equation: . g(X1 , . . . , Xn ) = g(f (X0 , X0 ), f (X1 , X1 ), . . . , f (Xn−1 , Xn−1 )) where X0 , . . . , Xn are distinct. By case 1 this reduces to: . . . {X1 = f (X0 , X0 ), X2 = f (X1 , X1 ), . . . , Xn = f (Xn−1 , Xn−1 )} Assume that the equation selected in step i is of the form Xi = f (. . . , . . .). Then in the . k-th iteration the selected equation is of the form Xk = Tk where Ti+1 := f (Ti , Ti ) and T0 := X0 . Hence, |Ti+1 | = 2|Ti | + 1. That is, |Tn | > 2n . This shows the exponential dependency of the unification time on the length of the structures. In this example the growth of the argument lengths is caused by duplication of subterms. As a matter of fact, the same check is repeated many times. Something that could be avoided by sharing various instances of the same structure. In the literature one can find linear algorithms but they are sometimes quite elaborate. On the other hand, Prolog systems usually “solve” the problem simply by omitting the occur-check during unification. Roughly speaking such an approach corresponds to a solved form algorithm where case 5a–b is replaced by: 1 The
size of a term is the total number of constant, variable and functor occurrences in t.
42
Chapter 3: SLD-Resolution . X = t where X 6= t and X has more than one occurrence in E ⇒ replace all other occurrences of X by t; % case 5
A pragmatic justification for this solution is the fact that rule 5a (occur check) never is used during the computation of many Prolog programs. There are sufficient conditions which guarantee this, but in general this property is undecidable. The ISO Prolog standard (1995) states that the result of unification is undefined if case 5b can be applied to the set of equations. Strictly speaking, removing case 5a causes looping of the algorithm on equations where case 5a would otherwise apply. For example, an . attempt to solve X = f (X) by the modified algorithm will produce a new equation . . X = f (f (X)). However, case 5 is once again applicable yielding X = f (f (f (f (X)))) and so forth. In practice many Prolog systems do not loop, but simply bind X to the infinite structure f (f (f (. . .))). (The notation X/f (∞) will be used to denote this binding.) Clearly, {X/f (∞)} is an infinite “unifier” of X and f (X). It can easily be represented in the computer by a finite cyclic data structure. But this amounts to generalization of the concepts of term, substitution and unifier for the infinite case not treated in classical logic. Implementation of unification without occur-check may result in unsoundness as will be illustrated in Example 3.21. Before concluding the discussion about unification we study the notion of most general unifier in more detail. It turns out that the notion of mgu is a subtle one; For instance, there is generally not a unique most general unifier of two terms s and t. A trivial . example is the equation f (X) = f (Y ) which has at least two mgu’s; namely {X/Y } and {Y /X}. Part of the confusion stems from the fact that (“being more general than”) is not an ordering relation. It is reflexive: That is, any substitution θ is “more general” than itself since θ = θ. As might be expected it is also transitive: If θ1 = θ2 ω1 and θ2 = θ3 ω2 then obviously θ1 = θ3 ω2 ω1 . However, is not anti-symmetric. For instance, consider the substitution θ := {X/Y, Y /X} and the identity substitution . The latter is obviously more general than θ since θ = θ. But θ is also more general than , since = θθ. It may seem odd that two distinct substitutions are more general than one another. Still there is a rational explanation. First consider the following definition: Definition 3.9 (Renaming) A substitution {X1 /Y1 , . . . , Xn /Yn } is called a renaming substitution iff Y1 , . . . , Yn is a permutation of X1 , . . . , Xn . A renaming substitution represents a bijective mapping between variables (or more generally terms). Such a substitution always preserves the structure of a term; if θ is a renaming and t a term, then tθ and t are equivalent but for the names of the variables. Now, the fact that a renaming represents a bijection implies that there must be an inverse mapping. Indeed, if {X1 /Y1 , . . . , Xn /Yn } is a renaming then {Y1 /X1 , . . . , Yn /Xn } is its inverse. We denote the inverse of θ by θ−1 and observe that θθ−1 = θ−1 θ = . Proposition 3.10 Let θ be an mgu of s and t and assume that ω is a renaming. Then θω is an mgu of s and t. The proof of the proposition is left as an exercise. So is the proof of the following proposition:
3.3 SLD-Resolution
43
Proposition 3.11 Let θ and σ be substitutions. If θ σ and σ θ then there exists a renaming substitution ω such that σ = θω (and θ = σω −1 ). Thus, according to the above propositions, the set of all mgu’s of two terms is closed under renaming.
3.3
SLD-Resolution
The method of reasoning discussed informally in Section 3.1 can be summarized as the following inference rule: ∀ ¬ (A1 ∧ · · · ∧ Ai−1 ∧ Ai ∧ Ai+1 ∧ · · · ∧ Am ) ∀ (B0 ← B1 ∧ · · · ∧ Bn ) ∀ ¬ (A1 ∧ · · · ∧ Ai−1 ∧ B1 ∧ · · · ∧ Bn ∧ Ai+1 ∧ · · · ∧ Am )θ or (using logic programming notation): ← A1 , . . . , Ai−1 , Ai , Ai+1 , . . . , Am B0 ← B1 , . . . , Bn ← (A1 , . . . , Ai−1 , B1 , . . . , Bn , Ai+1 , . . . , Am )θ where (i ) A1 , . . . , Am are atomic formulas; (ii ) B0 ← B1 , . . . , Bn is a (renamed) definite clause in P (n ≥ 0); (iii ) mgu(Ai , B0 ) = θ. The rule has two premises — a goal clause and a definite clause. Notice that each of them is separately universally quantified. Thus the scopes of the quantifiers are disjoint. On the other hand, there is only one universal quantifier in the conclusion of the rule. Therefore it is required that the sets of variables in the premises are disjoint. Since all variables of the premises are bound it is always possible to rename the variables of the definite clause to satisfy this requirement (that is, to apply some renaming substitution to it). The goal clause may include several atomic formulas which unify with the head of some clause in the program. In this case it may be desirable to introduce some deterministic choice of the selected atom Ai for unification. In what follows it is assumed that this is given by some function which for a given goal selects the subgoal for unification. The function is called the selection function or the computation rule. It is sometimes desirable to generalize this concept so that, in one situation, the computation rule selects one subgoal from a goal G but, in another situation, selects another subgoal from G. In that case the computation rule is not a function on goals but something more complicated. However, for the purpose of this book this extra generality is not needed. The inference rule presented above is the only one needed for definite programs. It is a version of the inference rule called the resolution principle, which was introduced by J. A. Robinson in 1965. The resolution principle applies to clauses. Since definite clauses are restricted clauses the corresponding restricted form of resolution presented below is called SLD-resolution (Linear resolution for Definite clauses with Selection function).
44
Chapter 3: SLD-Resolution
Next the use of the SLD-resolution principle is discussed for a given definite program P . The starting point, as exemplified in Section 3.1, is a definite goal clause G0 of the form: ← A1 , . . . , Am
(m ≥ 0)
From this goal a subgoal Ai is selected (if possible) by the computation rule. A new goal clause G1 is constructed by selecting (if possible) some renamed program clause B0 ← B1 , . . . , Bn (n ≥ 0) whose head unifies with Ai (resulting in an mgu θ1 ). If so, G1 will be of the form: ← (A1 , . . . , Ai−1 , B1 , . . . , Bn , Ai+1 , . . . , Am )θ1 (According to the requirement above, the variables of the program clause are being renamed so that they are different from those of G0 .) Now it is possible to apply the resolution principle to G1 thus obtaining G2 , etc. This process may or may not terminate. There are two cases when it is not possible to obtain Gi+1 from Gi : • the first is when the selected subgoal cannot be resolved (i.e. is not unifiable) with the head of any program clause; • the other case appears when Gi = 2 (i.e. the empty goal). The process described above results in a finite or infinite sequence of goals starting with the initial goal. At every step a program clause (with renamed variables) is used to resolve the subgoal selected by the computation rule < and an mgu is created. Thus, the full record of a reasoning step would be a pair hGi , Ci i, i ≥ 0, where Gi is a goal and Ci a program clause with renamed variables. Clearly, the computation rule < together with Gi and Ci determines (up to renaming of variables) the mgu (to be denoted θi+1 ) produced at the (i + 1)-th step of the process. A goal Gi+1 is said to be derived (directly) from Gi and Ci via < (or alternatively, Gi and Ci resolve into Gi+1 ). Definition 3.12 (SLD-derivation) Let G0 be a definite goal, P a definite program and < a computation rule. An SLD-derivation of G0 (using P and ’/2 separates the left- and right-hand sides of a production rule;
10.4 Definite Clause Grammars (DCGs)
173
• the empty string is denoted by the empty list. This means that DCGs can be represented as terms. For instance, the rule: np(X) → the noun(X) will be written as the term: np(X) --> [the], noun(X) or using standard syntax: ’-->’(np(X), ’,’([the], noun(X))) In addition Prolog often allows special treatment of nonterminals which always derive the empty string. Consider the language where each string consists of an even number of a’s followed by the same number of b’s in turn followed by the same number of c’s. This language can be specified as follows using DCGs with Prolog syntax: abc a(0) a(s(N )) b(0) b(s(N )) c(0) c(s(N )) even(0) even(s(s(N )))
--> --> --> --> --> --> --> --> -->
a(N ), b(N ), c(N ), even(N ). [ ]. [a], a(N ). [ ]. [b], b(N ). [ ]. [c], c(N ). [ ]. even(N ).
In this example the occurrence of even(N ) in the first rule always derives the empty string. In this respect it can be removed from the rule. However, the primary function of this nonterminal is to constrain bindings of N to terms representing even numbers. That is, it not only defines the language consisting solely of the empty string but also defines a relation (in this case the property of being an even natural number). To distinguish such nonterminals from those which derive nonempty strings they are written within curly brackets, and the definition of the relation is written directly in Prolog. Thus, in Prolog it is possible to write the rule: abc --> a(N ), b(N ), c(N ), {even(N )}. together with the definite program: even(0). even(s(s(N ))) ← even(N ). replacing the first and the two final production rules of the previous DCG. Now since calls to Prolog may be inserted into a DCG it is also possible to utilize the built-in predicates of Prolog in a DCG. Example 10.10 This idea is illustrated by the following example, which is a grammar that recognizes arithmetic expressions but, more than that, also computes the value of the expression:
174
Chapter 10: Logic and Grammars expr(X) expr(X) expr(X) term(X) term(X) term(X) factor (X)
--> --> --> --> --> --> -->
term(Y ), [+], expr(Z), {X is Y + Z}. term(Y ), [−], expr(Z), {X is Y − Z}. term(X). factor (Y ), [ ∗ ], term(Z), {X is Y ∗ Z}. factor (Y ), [ / ], term(Z), {X is Y /Z}. factor (X). [X], {integer(X)}.
For instance, the last rule states that any string which consists of a single terminal which is an integer is a factor with the same value as the terminal. Similarly, the first rule states that any string which starts with a term of value Y followed by “+” followed by an expression of value Z is an expression of value Y + Z. As already discussed, when a DCG is loaded into a Prolog system it is usually compiled into a Prolog program similar in style to those in Section 10.2. This transformation is quite simple and it is discussed in the next section. Some implementations of Prolog do not include this feature. Fortunately, it is not very hard to write an interpreter for DCGs similar to the Prolog-interpreter in Example 8.7. For this purpose, view a DCG-rule as a binary fact with predicate symbol ’-->’/2 where the first and second arguments consist of the left- and right-side of the production rule. The relationship between strings of terminals/nonterminals and the derivable terminal strings can then be defined as follows: derives([ ], S − S). derives([X], [X|S] − S). derives({X}, S − S) ← call(X). derives((X, Y ), S0 − S2 ) ← derives(X, S0 − S1 ), derives(Y, S1 − S2 ). derives(X, S0 − S1 ) ← (X --> Y ), derives(Y, S0 − S1 ). The interpreter is surprisingly simple. Declaratively the clauses state the following: • the empty string derives itself. That is, the difference between S and S for any S; • the second clause says that the terminal string [X] derives itself. That is, the difference between [X|S] and S for any S; • a nonterminal X in curly brackets derives the empty string if the goal ← X has a refutation; • if the string X derives the terminal string S0 − S1 and the string Y derives the terminal string S1 −S2 then the string (X, Y ) (that is, Y appended to X) derives the terminal string S0 − S2 ; • if there is a rule (X --> Y ) such that Y derives the terminal string S0 − S1 then the nonterminal X derives the same terminal string.
10.5 Compilation of DCGs into Prolog
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For instance, in the presence of the DCG of Example 10.9 the goal: ← derives(sentence(X), [john, runs] − [ ]). succeeds with answer X = s(john, runs). Similarly the grammar in Example 10.10 and the goal: ← derives(expr(X), [2, +, 3, ∗, 4] − [ ]). result in the answer X = 14.
10.5
Compilation of DCGs into Prolog
The standard treatment of DCGs in most Prolog systems is to compile them directly into Prolog clauses. Since each production rule translates into one clause, the transformation is relatively simple. The clause obtained as a result of the transformations described below may differ slightly from what is obtained in some Prolog systems but the principle is the same. The general idea is the following — consider a production rule of the form: p(t1 , . . . , tn ) --> T1 , . . . , Tm Assume that X0 , . . . , Xm are distinct variables which do not appear in the rule. Then the production rule translates into the definite clause: p(t1 , . . . , tn , X0 , Xm ) ← A1 , . . . , Am where: • if Ti is of the form q(s1 , . . . , sj ), then Ai is q(s1 , . . . , sj , Xi−1 , Xi ); • if Ti is of the form [T ], then Ai is connects(Xi−1 , T, Xi );2 . • if Ti is of the form {T }, then Ai is T, Xi−1 = Xi ; . • if Ti is of the form [ ], then Ai is Xi−1 = Xi . For instance, the first rule of Example 10.10 is transformed as follows: expr(X) --> expr(X, X0 , X4 ) ← term(Y ), term(Y, X0 , X1 ), [+], ⇒ connects(X1 , +, X2 ), expr(Z), expr(Z, X2 , X3 ), . {X is Y + Z}. X is Y + Z, X3 = X4 . Some simplifications can be made to the final result — in particular, subgoals of the . form X = Y may be omitted if all occurrences of Y are replaced by the variable X. This means that the result obtained above can be simplified into: 2 Many Prolog systems use instead a built-in predicate ’C’(A, B, C) with the same semantics as connects(A, B, C).
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Chapter 10: Logic and Grammars expr(X, X0 , X3 ) ← term(Y, X0 , X1 ), connects(X1 , +, X2 ), expr(Z, X2 , X3 ), X is Y + Z.
When all rules are translated in this way and the program is extended by the definition connects([X|Y ], X, Y ), the resulting program can be used to refute goals like ← expr(X, [2, +, 3, ∗, 4], [ ]), with the expected answer X = 14. A CFG (which is a special case of a DCG) like the one in Example 10.1 translates into the program of Example 10.5 except that arguments of the form X − Y are split into two arguments and that the names of the variables may differ. Example 10.11 As a final example the translation of Example 10.9 results in the following program: sentence(s(X, Y ), X0 , X2 ) ← np(X, N, X0 , X1 ), vp(Y, N, X1 , X2 ). np(john, singular(3), X0, X1 ) ← connects(X0 , john, X1 ). np(they, plural(3), X0, X1 ) ← connects(X0 , they, X1 ). vp(run, plural(X), X0 , X1 ) ← connects(X0 , run, X1 ). vp(runs, singular(3), X0 , X1 ) ← connects(X0 , runs, X1 ). connects([X|Y ], X, Y ). Given the goal ← sentence(X, [john, runs], [ ]) Prolog replies with the answer X = s(john, runs).
Exercises 10.1 Write a DCG which describes the language of strings of octal numbers. Extend the grammar so that the decimal value of the string is returned. For instance, the goal ← octal(X, [4, 6], [ ]) should succeed with X = 38. 10.2 Write a DCG which accepts strings in the language am bn cm dn , (n, m ≥ 0). 10.3 Consider the following CFG: hbleati → b haaai haaai → a haaai → a haaai Describe the same language using DCG notation. Then “compile” the specification into a Prolog program and write an SLD-refutation which proves that the string “b a a” is in the language of hbleati.
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10.4 Explain the usage of the following DCG: --> [ ]. x([ ], X, X) x([X|Y ], Z, [X|W ]) --> x(Y, Z, W ). 10.5 Write an interface to a database that facilitates communication in natural language. 10.6 Define a concrete syntax for the imperative language outlined in exercise 8.6. Then write a compiler which translates the concrete syntax (i.e. strings of characters) into the given abstract syntax. It is probably a good idea to split the translation into two phases. In the first phase the string is translated into a list of lexical items representing identifiers, reserved words, operators etc, after which the string is translated into the abstract syntax. The following context-free grammar may serve as a starting point for the compiler: hcmnd i → | | | | hbool i → | hexpr i → | | hvar i → hnati → hopi →
skip hvar i := hexpr i if hbool i then hcmnd i else hcmnd i fi while hbool i do hcmnd i od hcmnd i ; hcmnd i hexpr i > hexpr i ... hvar i hnati hexpr i hopi hexpr i x | y | z | ... 0 | 1 | 2 | ... + | − | ∗ | ...
Hint: Most Prolog systems permit using the syntax ”prolog” as an alternative for the list of ASCII-characters [112, 114, 111, 108, 111, 103].
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Chapter 11 Searching in a State-space
11.1
State-spaces and State-transitions
Many problems in computer science can be formulated as a possibly infinite set S of states and a binary transition-relation over this state-space. Given some start-state s0 ∈ S and a set G ⊆ S of goal-states such problems consist in determining whether there exists a sequence: s0
s1 , s1
s2 , s2
s3 , · · · sn−1
sn
such that sn ∈ G (i.e. to determine if hs0 , sn i is in the transitive and reflexive closure, ∗ , of ). More informally the states can be seen as nodes in a graph whose edges represent the pairs in the transition-relation. Then the problem reduces to that of finding a path from the start-state to one of the goal-states. Example 6.7 embodies an instance of such a problem — the state-space consisted of a finite set of states named by a, b, c, d, e, f and g. The predicate symbol edge/2 was used to describe the transition relation and path/2 described the transitive and reflexive closure of the transition relation. Hence, the existence of a path from, for instance, the state a to e is checked by giving the goal ← path(a, e). Now this is by no means the only example of such a problem. The following ones are all examples of similar problems: • Planning amounts to finding a sequence of worlds where the initial world is transformed into some desired final world. For instance, the initial world may consist of a robot and some parts. The objective is to find a world where the parts are assembled in some desirable way. Here the description of the world is a state and the transformations which transform one world to another can be seen as a transition relation. • The derivation of a string of terminals α from a nonterminal A can also be viewed 179
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Chapter 11: Searching in a State-space in this way. The state-space consists of all strings of terminals/nonterminals. The string A is the start-state and α the goal-state. The relation “⇒” is the transition relation and the problem amounts to finding a sequence of derivation steps A ⇒ · · · ⇒ α.
• Also SLD-derivations may be formulated in this way — the states are goals and the transition relation consists of the SLD-resolution principle which produces a goal Gi+1 out of another goal Gi and some program clause Ci . In most cases the start-state is the initial goal and the goal-state is the empty goal. Consider the following two clauses of Example 6.7 again: path(X, X). path(X, Z) ← edge(X, Y ), path(Y, Z). Operationally the second clause reads as follows provided that Prolog’s computation rule is employed — “To find a path from X to Z, first find an edge from X to Y and then find a path from Y to Z”. That is, first try to find a node adjacent to the start-state, and then try to find a path from the new node to the goal-state. In other words, the search proceeds in a forward direction — from the start-state to the goalstate. However, it is easy to modify the program to search in the opposite direction assuming that Prolog’s computation rule is used — simply rewrite the second clause as: path(X, Z) ← edge(Y, Z), path(X, Y ). The decision whether to search in a forward or backward direction depends on what the search space looks like. Such considerations will not be discussed here, but the reader is referred to the AI-literature The path/2-program above does not work without modifications if there is more than one goal-state. For instance, if both f and g are goal-states one has to give two goals. An alternative solution is to extend the program with the property of being a goal-state: goal state(f ). goal state(g). Now the problem of finding a path from a to one of the goal-states reduces to finding a refutation of the goal: ← path(a, X), goal state(X). The program above can be simplified if the goal-state is known in advance. In this case it is not necessary to use the second argument of path/2. Instead the program may be simplified into:
p q
p q
path( goal ). path(X) ← edge(X, Y ), path(Y ).
where goal is a term representing the goal-state (if there are several goal-states there will be one such fact for each state).
11.2 Loop Detection
11.2
181
Loop Detection
One problem mentioned in connection with Example 6.7, appears when the graph defined by the transition relation is cyclic. Example 11.1 Consider the program: path(X, X). path(X, Z) ← edge(X, Y ), path(Y, Z). edge(a, b). edge(b, a). edge(a, c). edge(b, d). edge(b, e). edge(c, e). edge(d, f ). edge(e, f ). edge(e, g). As pointed out in Chapter 6 the program may go into an infinite loop for certain goals — from state a it is possible to go to state b and from this state it is possible to go back to state a via the cycle in the transition relation. One simple solution to such problems is to keep a log of all states already visited. Before moving to a new state it should be checked that the new state has not already been visited. Example 11.2 The following program extends Example 11.1 with a log: path(X, Y ) ← path(X, Y, [X]). path(X, X, Visited). path(X, Z, Visited ) ← edge(X, Y ), not member(Y, Visited ), path(Y, Z, [Y |Visited ]). member(X, [X|Y ]). member(X, [Y |Z]) ← member(X, Z). Declaratively the recursive clause of path/3 says that — “there is a path from X to Z if there is an edge from X to Y and a path from Y to Z such that Y has not already been visited”. At first glance the solution may look a bit inelegant and there certainly are more sophisticated solutions around. However, carrying the log around is not such a bad idea after all — in many problems similar to the one above, it is not sufficient just to answer “yes” or “no” to the question of whether there is a path between two states. Often it is necessary that the actual path is returned as an answer to the goal. As an example, it is not much use to know that there is a plan which assembles some pieces of material into a gadget; in general one wants to see the actual plan. Example 11.3 This extension can be implemented through the following modification of Example 11.2:
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Chapter 11: Searching in a State-space path(X, Y, P ath) ← path(X, Y, [X], P ath). path(X, X, Visited, Visited ). path(X, Z, Visited, P ath) ← edge(X, Y ), not member(Y, Visited), path(Y, Z, [Y |Visited ], P ath).
With these modifications the goal ← path(a, d, X) succeeds with the answer X = [d, b, a] which says that the path from a to d goes via the intermediate state b. Intuitively, path(A, B, C, D) can be interpreted as follows — “The difference between D and C constitutes a path from A to B”.
11.3
Water-jug Problem (Extended Example)
The discussion above will be illustrated with the well-known water-jug problem often encountered in the AI-literature. The problem is formulated as follows: Two water jugs are given, a 4-gallon and a 3-gallon jug. Neither of them has any type of marking on it. There is an infinite supply of water (a tap?) nearby. How can you get exactly 2 gallons of water into the 4-gallon jug? Initially both jugs are empty. The problem can obviously be described as a state-space traversal — a state is described by a pair hx, yi where x represents the amount of water in the 4-gallon jug and y represents the amount of water in the 3-gallon jug. The start-state then is h0, 0i and the goal-state is any pair where the first component equals 2. First of all some transformations between states must be formulated. The following is by no means a complete set of transformations but it turns out that there is no need for additional ones: • empty the 4-gallon jug if it is not already empty; • empty the 3-gallon jug if it is not already empty; • fill up the 4-gallon jug if it is not already full; • fill up the 3-gallon jug if it is not already full; • if there is enough water in the 3-gallon jug, use it to fill up the 4-gallon jug until it is full; • if there is enough water in the 4-gallon jug, use it to fill up the 3-gallon jug until it is full; • if there is room in the 4-gallon jug, pour all water from the 3-gallon jug into it; • if there is room in the 3-gallon jug, pour all water from the 4-gallon jug into it.
11.4 Blocks World (Extended Example)
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It is now possible to express these actions as a binary relation between two states. The binary functor :/2 (written in infix notation) is used to represent a pair: action(X action(X action(X action(X action(X action(X action(X action(X
: Y, 0 : Y ) ← X > 0. : Y, X : 0) ← Y > 0. : Y, 4 : Y ) ← X < 4. : Y, X : 3) ← Y < 3. : Y, 4 : Z) ← X < 4, Z is Y − (4 − X), Z ≥ 0. : Y, Z : 3) ← Y < 3, Z is X − (3 − Y ), Z ≥ 0. : Y, Z : 0) ← Y > 0, Z is X + Y, Z ≤ 4. : Y, 0 : Z) ← X > 0, Z is X + Y, Z ≤ 3.
The definition of a path is based on the program in Example 11.3. However, since the goal-state is known to be h2, Xi for any value of X (or at least 0 ≤ X ≤ 3) there is no need for the second argument of path/4. With some minor additional changes the final version looks as follows: path(X) ← path(0 : 0, [0 : 0], X). path(2 : X, Visited , Visited ). path(State, Visited, Path) ← action(State, NewState), not member(NewState, Visited ), path(NewState, [NewState|Visited ], Path). member(X, [X|Y ]). member(X, [Y |Z]) ← member(X, Z). Given this program and the goal ← path(X) several answers are obtained some of which are rather naive. One answer is X = [2 : 0, 0 : 2, 4 : 2, 3 : 3, 3 : 0, 0 : 3, 0 : 0]. That is, first fill the 3-gallon jug and pour this water into the 4-gallon jug. Then the 3gallon jug is filled again, and the 4-gallon jug is filled with water from the 3-gallon jug. The last actions are to empty the 4-gallon jug and then pour the content of the 3-gallon jug into it. Another answer is X = [2 : 0, 0 : 2, 4 : 2, 3 : 3, 3 : 0, 0 : 3, 4 : 3, 4 : 0, 0 : 0]. In all 27 answers are produced.
11.4
Blocks World (Extended Example)
A similar problem is the so-called blocks world. Consider a table with three distinct positions. On the table are a number of blocks which may be stacked on top of each other. The aim is to move the blocks from a given start-state to a goal-state. Only blocks which are free (that is, with no other block on top of them) can be moved. The first step is to determine how to represent the state. A reasonable solution is to use a ternary functor state/3 to represent the three positions of the table. Furthermore, use the constant table to denote the table. Finally represent by on(X, Y ) the fact that X is positioned on top of Y . That is, state(on(c, on(b, on(a, table))), table, table) represents the state:
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c b a 1
2
3
The following are all possible actions that transform the state: • if the first position is nonempty the topmost block can be moved to either the second or the third position; • if the second position is nonempty the topmost block can be moved to either the first or the third position; • if the third position is nonempty the topmost block can be moved to either the first or the second position. The first action may be formalized as follows: move(state(on(X, New X), OldY, Z), state(NewX, on(X, OldY ), Z)). move(state(on(X, New X), Y, OldZ), state(NewX, Y, on(X, OldZ))). The remaining two actions may be formalized in a similar way. Finally the program is completed by adding the path-program from Example 11.3 (where edge/2 is renamed into move/2). It is now possible to find the path from the start-state above to the following goal-state: c a b 1
2
3
by giving the goal: ← path( state(on(c, on(b, on(a, table))), table, table), state(table, table, on(c, on(a, on(b, table)))), X). One answer to the goal is: X = [ state(table, table, on(c, on(a, on(b, table)))), state(table, on(c, table), on(a, on(b, table))), state(on(a, table), on(c, table), on(b, table)), state(on(b, on(a, table)), on(c, table), table), state(on(c, on(b, on(a, table))), table, table) ] That is, first move c to position 2. Then move b to position 3 and a on top of b. Finally move c on top of a.
11.5 Alternative Search Strategies
11.5
185
Alternative Search Strategies
For many problems the depth-first traversal of a state-space is sufficient as shown above — the depth-first strategy is relatively simple to implement and the memory requirements are relatively modest. However, sometimes there is need for alternative search strategies — the depth-first traversal may be stuck on an infinite path in the state-space although there are finite paths which lead to (one of) the goal-states. Even worse, sometimes the branching of the state-space is so huge that it is simply not feasible to try all possible paths — instead one has to rely on heuristic knowledge to reduce the number of potential paths. When describing such problems by means of logic programming there are two solutions to this problem — either the logic program (for instance that in Example 11.1) is given to an inference system which employs the desired search strategy; or one writes a Prolog program which solves the problem using the desired strategy (however, this program is of course executed using the standard depth-first technique of Prolog). In this section an example of a Prolog program which searches a (simple) tree using a breadth-first traversal is shown. Example 11.4 Consider the following tree: a
,,@@ b c ,,@@
d
e
,,@@
g
f
h
To look for a path in a tree (or a graph) using a breadth-first strategy means first looking at all paths of length 1 from the start-state (or to the goal-state). Then all paths of length 2 are investigated. The process is repeated until a complete path from the start- to a goal-state is found. The tree above will be represented using a binary predicate symbol children/2 where the first argument is the name of a node of the tree and the second argument is the names of the children of that node. Hence the tree above is represented as follows: children(a, [b, c]). children(b, [d, e]). children(c, [f ]). children(e, [g, h]). In order to realize the breadth-first strategy paths will be represented by “reversed” lists of nodes. For instance, [d, b, a] represents the path which starts at the root of the tree and proceeds to d via the intermediate node b. Given all paths of length n from a start-state the problem of finding a path to a goal-state reduces to finding a path from the end of one of these paths to the goal-state. Initially this amounts to finding a path from the empty branch [X] (where X is the start-state) to a goal-state. That is:
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Chapter 11: Searching in a State-space path(X, Y ) ← bf path([[X]], Y ).
The first argument of bf path/2 consists of a collection of paths (represented by a list) and the second argument is the goal-state (this may easily be generalized to several goal-states): bf path([ [Leaf |Branch] | Branches ], Leaf ). bf path([ [Leaf |Branch] | Branches ], Goal) ← children(Leaf , Adjacent), expand([Leaf |Branch], Adjacent, Expanded), append(Branches, Expanded, NewBranches), bf path(NewBranches , Goal). bf path([ [Leaf |Branch] | Branches ], Goal) ← not children(Leaf , Leaves), bf path(Branches, Goal). The last clause exploits unsafe use of negation and applies when a path cannot be expanded any further. Notice, that in order to implement a breadth-first search, it is vital that Expanded is appended to Branches. The other way around would lead to a depth-first search. Thus, the first argument of bf path/2 behaves as a FIFO-queue where a prefix contains paths of length n and where the rest of the queue contains paths of length n + 1. expand/3 describes the relation between a path X, the children X1 , . . . , Xn of the final node in X and the paths obtained by adding X1 , X2 , etc. to the end of X: expand(X, [ ], [ ]). expand(X, [Y |Z], [ [Y |X] | W ]) ← expand(X, Z, W ). For instance, the goal ← expand([b, a], [d, e], X) succeeds with the answer X = [[d, b, a], [e, b, a]]. When extended with the usual definition of append/3 the goal ← path(a, X) yields all eight possible solutions. That is, a, b, c, d, e, f , g and h.
Exercises 11.1 A chessboard of size N × N is given — the problem is to move a knight across the board in such a way that every square on the board is visited exactly once. The knight may move only in accordance with the standard chess rules. 11.2 A farmer, a wolf and a goat are standing on the same river-bank accompanied by a cabbage-head (a huge one!). A boat is available for transportation. Unfortunately, it has room for only two individuals including the cabbage-head. To complicate things even more (1) the boat can be operated only by the farmer and (2) if the goat is left alone with the wolf it will be eaten. Similarly if the cabbage-head is left alone with the goat. Is there some way for them to cross the river without anyone being eaten? 11.3 Three missionaries and three cannibals are standing on the same side of a river. A boat with room for two persons is available. If the missionaries on
Exercises
187
either side of the river are outnumbered by cannibals they will be done away with. Is there some way for all missionaries and cannibals to cross the river without anyone being eaten? First solve the problem using a depth-first search strategy. (In which case a log must be used to prune infinite paths.) Then solve the same problem using a breadth-first strategy similar to that on page 186. 11.4 (Towers of Hanoi) Three pins are available together with N disks of different sizes. Initially all disks are stacked (smaller on top of bigger) on the leftmost pin. The task is to move all disks to the rightmost pin. However, at no time may a disk be on top of a smaller one. Hint: there is a very simple and efficient algorithmic solution to this puzzle. However, it may also be solved with the techniques described above. 11.5 How can the program in Example 11.4 be modified to avoid the use of negation?
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PART III ALTERNATIVE LOGIC PROGRAMMING SCHEMES
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Chapter 12 Logic Programming and Concurrency
12.1
Algorithm = Logic + Control
The construction of a computer program can be divided into two phases which are usually intertwined — the formulation of the actual problem (what the problem is) and the description of how to solve the problem. Together they constitue an algorithm. This idea is the heart of logic programming — the logic provides a description of the problem and SLD-resolution provides the means for executing the description. However, the logic has a meaning in itself — its declarative semantics — which is independent of any particular execution strategy. This means that, as long as the inference mechanism is sound, the behaviour of the algorithm may be altered by selecting an alternative inference mechanism. We do not have to go as far as abandoning SLD-resolution — the behaviour of the execution can be altered simply by choosing different computation rules as illustrated by the following example: Example 12.1 The following execution trace illustrates the impact of a more versatile computation rule than the one used in Prolog. First consider the program: (1) append([ ], X, X). (2) append([X|Y ], Z, [X|W ]) ← append(Y, Z, W ). (3) succlist([ ], [ ]). (4) succlist([X|Y ], [Z|W ]) ← succlist(Y, W ), Z is X + 1. Then consider the following SLD-derivation whose initial goal consists of two components (subgoals). Each of the subsequent goals in the derivation can be divided into two halves originating from the components of the initial goal as visualized by the 191
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frames: ← append([4, 5], [3], X) succlist(X, Res)
(G0 )
Resolving append([4, 5], [3], X) using (2) yields the binding [4|W0 ] for X and the new goal: ← append([5], [3], W0 )
succlist([4|W0], Res)
(G1 )
Resolving succlist([4|W0 ], Res) using (4) yields the binding [Z1 |W1 ] for Res and the goal: ← append([5], [3], W0 )
succlist(W0 , W1 ), Z1 is 4 + 1
(G2 )
Resolving append([5], [3], W0 ) using (2) yields the binding [5|W2 ] for W0 : ← append([ ], [3], W2 )
succlist([5|W2], W1 ), Z1 is 4 + 1
(G3 )
Selection of Z1 is 4 + 1 binds Z1 to 5. Consequently Res is bound to [5|W1 ]: ← append([ ], [3], W2 )
succlist([5|W2], W1 )
(G4 )
Resolving succlist([5|W2 ], W1 ) using (4) yields the binding [Z4 |W4 ] for W1 : ← append([ ], [3], W2 )
succlist(W2 , W4 ), Z4 is 5 + 1
(G5 )
Resolving append([ ], [3], W2 ) using (1) binds W2 to [3]: ←
succlist([3], W4), Z4 is 5 + 1
(G6 )
Selection of Z4 is 5 + 1 binds Z4 to 6 and Res is bound to [5, 6|W4 ]: ←
succlist([3], W4)
(G7 )
In the next step W4 is bound to [Z7 |W7 ] yielding: ←
succlist([ ], W7 ), Z7 is 3 + 1
(G8 )
Selection of succlist([ ], W7 ) binds W7 to [ ]: ←
Z7 is 3 + 1
(G9 )
Finally Z7 is bound to 4 yielding a refutation where the binding for Res is the list [5, 6, 4]. Thus, 10 SLD-steps are needed to refute the initial goal. Notice that it is not possible to improve on this by choosing an alternative computation rule — no matter what rule is used, the refutation will have the length 10.
12.2 And-parallelism
[3] -
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[4, 5]
append
[4, 5, 3]
-
-
[5, 6, 4] succlist
Figure 12.1: Process-interpretation of G0
With this versatile computation rule the two subgoals in G0 may be viewed as two processes which communicate with each other using the (shared) variable X. The frames in the derivation capture the internal behaviour of the processes and the shared variable acts as a “communication channel” where a stream of data flows — namely the elements of the list to which the shared variable X is (incrementally) bound to (first 4 then 5 and finally 3). See Figure 12.1. Notice that there is no real parallelism in this example. The executions of the two processes are only interleaved with each other. The control is merely shifted between them and there is no real gain in performance. This type of control is commonly known as coroutining. The possibility of viewing subgoals as processes and goals as nets of communicating processes connected by means of shared variables implies yet another interpretation of logic programs in addition to its operational and declarative meaning. This new view of logic programming extends the possible application areas of logic programming to include also process programming (like operating systems, simulators or industrial process control systems).
12.2
And-parallelism
Instead of solving the subgoals in a goal in sequence (using SLD-resolution) it is possible to use an operational semantics where some of the subgoals are solved in parallel. This is commonly called AND-parallelism. However, since the subgoals may contain shared variables it is not always feasible to solve all of them independently. Consider the following program: do this(a). do that(b). A goal of the form: ← do this(X), do that(X). would fail using SLD-resolution. However, the two subgoals are solvable separately. The leftmost subgoal binds X to a and the rightmost binds X to b. When two subgoals contain a shared variable special care must be taken so that different occurrences of the variable do not get bound to inconsistent values. This calls for some form of communication/synchronization between the subgoals. However there are some special cases when two or more derivation-steps can be carried out independently: • when the subgoals have no shared variable, and
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• when at most one subgoal binds each shared variable. Consider the derivation in Example 12.1 again. Note that both subgoals in G1 can be resolved in parallel since the shared variable (W0 ) is bound only by append([5], [3], W0 ). Similarly, all subgoals in G3 , G6 and G8 may be resolved in parallel. Thus, by exploiting AND-parallelism the goal may be solved in only five steps (G0 , G1 , G3 , G6 , G8 , G10 ) reducing the (theoretical) time of execution by 50%.
12.3
Producers and Consumers
One point worth noticing about Example 12.1 is that the execution is completely determinate — no selected subgoal unifies with more than one clause-head. However, this is not necessarily the case if some other computation rule is employed. For instance, the rightmost subgoal in G0 unifies with two different clauses. To reduce the search space it is desirable to have a computation rule which is “as determinate as possible”. Unfortunately it is rather difficult (if at all possible) to implement a computation rule which always selects a determinate subgoal. However, the programmer often has some idea how the program should be executed to obtain good efficiency (although not always optimal). Hence the programmer may be allowed to provide additional information describing how the program should be executed. The subgoal append([4, 5], [3], X) in G0 may be viewed as a process which consumes input from two streams ([4, 5] and [3]) and acts as a producer of bindings for the variable X. Similarly succlist(X, Res) may be viewed as a process which consumes bindings for the variable X and produces a stream of output for the variable Res. Since, in general, it is not obvious which subgoals are intended to act as consumers and producers of shared variable-occurrences the user normally has to provide a declaration. For instance, that the first two arguments of append/3 act as consumers and the third as producer. There are several ways to provide such information. In what follows we will use a notion of read-only variable which very closely resembles that employed in languages such as Concurrent Prolog (one of the first and most influential languages based on a concurrent execution model). Roughly speaking, each clause (including the goal) may contain several occurrences of a variable. On the other hand, variables can be bound at most once in an SLDderivation. This implies that at most one of the atoms in a clause acts as a producer of a binding for that variable whereas all remaining atoms containing some occurrence of the variable act as consumers. The idea employed in Concurrent Prolog is that the user annotates variable-occurrences appearing in consumer atoms by putting a question mark immediately after each occurrence. For instance, G0 may be written as follows: ← append([4, 5], [3], X), succlist(X?, Res). This means that the call to append/3 acts as a producer of values for X and that succlist/2 acts as a consumer of values for X and a producer of values for Res. Variables annotated by ’ ?’ are called read-only variables. Variables which are not annotated are said to be write-enabled. Now what is the meaning of a read-only variable? From a declarative point of view they are not different from write-enabled occurrences of the same variable. Consequently, the question-mark can be ignored in which case X and X? denote the same
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variable. However, from an operational point of view X and X? behave differently. The role of X? is to suspend unification temporarily if it is not possible to unify a subgoal with a clause head without producing a binding for X?. The unification can be resumed only when the variable X is bound to a non-variable by some other process containing a write-enabled occurrence of the variable. For instance, unification of p(X?) and p(f (Y )) suspends whereas unification of p(Y ?) and p(X) succeeds with mgu {X/Y ?}. Application of a substitution θ to a term or a formula is defined as before except that X?θ = (Xθ)?. Consequently, a read-only annotation may appear after nonvariable terms. In this case the annotation has no effect and can simply be removed. For instance: p(X?, Y, Z?){X/f (W ), Y /f (W ?)} = p(f (W )?, f (W ?), Z?) = p(f (W ), f (W ?), Z?) Example 12.2 Consider the following (nonterminating) program describing the producer-consumer problem with an unbounded buffer. That is, there is a producer which produces data and a consumer which consumes data and we require that the consumer does not attempt to consume data which is not there: producer([X|Y ]) ← get(X), producer(Y ). consumer([X|Y ]) ← print(X?), consumer(Y ?). We do not specify exctly how get/1 and print/1 are defined but only assume that the call get(X) suspends until some data (e.g. a text file) is available from the outside whereas print(X) is a printer-server which prints the file X. To avoid some technical problems we also assume that a call to producer/1 (resp. consumer/1) does not go ahead until get/1 (resp. print/1) succeeds. Now consider the goal: ← producer(X), consumer(X?). Because of the read-only annotation the second subgoal suspends. The first subgoal unifies with producer([X0 |Y0 ]) resulting in the mgu {X/[X0 |Y0 ]} and the new goal: ← get(X0 ), producer(Y0 ), consumer([X0 |Y0 ]). At this point only the third subgoal may proceed. (The first subgoal suspends until some external data becomes available and the second subgoal suspends until the first subgoal succeeds.) ← get(X0 ), producer(Y0 ), print(X0 ?), consumer(Y0 ?). This goal suspends until an external job arrives (to avoid having to consider how jobs are represented we just denote them by job n ). Assume that job 1 eventually arrives in which case get(X0 ) succeeds with X0 bound to job 1 : ← producer(Y0 ), print(job 1 ), consumer(Y0 ?). Then assume that the producer/1-process is reduced to: ← get(X1 ), producer(Y1 ), print(job 1 ), consumer([X1 |Y1 ]).
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and that a new job arrives from outside while the first is being printed: ← producer(Y1 ), print(job 1 ), consumer([job 2 |Y1 ]). The producer/1-process can now be reduced to: ← get(X2 ), producer(Y2 ), print(job 1 ), consumer([job 2 , X2 |Y2 ]). The whole goal suspends again until either (1) a new job arrives (in which case the new job is enqueued after the second job) or (2) printing of the first job ends (in which case the second job can be printed). As pointed out above the program does not terminate. Notice that it may happen that all subgoals in a goal become suspended forever. A trivial example is the goal ← consumer(X?). This situation is called deadlock.
12.4
Don’t Care Nondeterminism
The Prolog computation consists of a traversal of the SLD-tree. The branching of the tree occurs when the selected subgoal matches several clause heads. To be sure that no refutations are disregarded, a backtracking strategy is employed. Informally the system “does not know” how to obtain the answers so all possibilities are tried (unless, of course, the search gets stuck on some infinite branch). This is sometimes called “don’t know nondeterminism”. The traversal of the tree may be carried out in parallel. This is commonly called OR-parallelism. Notice that OR-parallelism does not necessarily speed up the discovery of a particular answer. Although full OR-parallelism may be combined with AND-parallelism this is seldom done because of implementation difficulties. Instead a form of limited ORparallelism is employed. The idea is to commit to a single clause as soon as possible when trying to solve a literal. Informally this means that all parallel attempts to solve a subgoal are immediately surrendered when the subgoal unifies with the head of some clause and certain subgoals in that clause are solved. To make this more precise the concept of commit operator is introduced. The commit operator divides the body of a clause into a guard- and body-part. The commit-operator may be viewed as a generalized cut operator in the sense that it cuts off all other attempts to solve a subgoal. This scheme is usually called “don’t care nondeterminism”. Intuitively this can be understood as follows — assume that a subgoal can be solved using several different clauses all of which lead to the same solution. Then it does not matter which clause to pick. Hence, it suffices to pick one of the clauses not caring about the others. Of course, in general it is not possible to tell whether all attempts will lead to the same solution and the responsibility has to be left to the user.
12.5
Concurrent Logic Programming
The concepts discussed above provide a basis for a class of programming languages based on logic programming. They are commonly called Concurrent Logic Programming languages or Committed Choice Languages. For the rest of this chapter the
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principles of these languages are discussed. To illustrate the principles we use a language similar to Shapiro’s Concurrent Prolog (1983a). By analogy to definite programs, the programs considered here are finite sets of guarded clauses. The general scheme of a guarded clause is as follows: H ← G1 , . . . , Gm | B1 , . . . , Bn
(m ≥ 0, n ≥ 0)
where H, G1 , . . . , Gm , B1 , . . . , Bn are atoms (possibly containing read-only annotations). H is called the head of the clause. G1 , . . . , Gm and B1 , . . . , Bn are called the guard and the body of the clause. The symbol “|” which divides the clause into a guardand body-part is called the commit operator. If the guard is empty the commit operator is not written out. To simplify the operational semantics of the language, guards are only allowed to contain certain predefined test-predicates — typically arithmetic comparisons. Such guards are usually called flat and the restriction of Concurrent Prolog which allows only flat guards is called Flat Concurrent Prolog (FCP). Like definite programs, FCP-programs are used to produce bindings for variables in goals given by the user. The initial goal is not allowed to contain any guard. Example 12.3 The following are two examples of FCP-programs for merging lists and deleting elements from lists: merge([ ], [ ], [ ]). merge([X|Y ], Z, [X|W ]) ← merge(Y ?, Z, W ). merge(X, [Y |Z], [Y |W ]) ← merge(X, Z?, W ). delete(X, [ ], [ ]). delete(X, [X|Y ], Z) ← delete(X, Y ?, Z). delete(X, [Y |Z], [Y |W ]) ← X 6= Y | delete(X, Z?, W ). Like definite clauses, guarded clauses have a logical reading: • all variables in a guarded clause are implicitly universally quantified — the readonly annotations have no logical meaning; • “←” denotes logical implication; • “|” and “,” denote logical conjunctions. Each clause of the program must contain exactly one commit operator (although usually not explicitly written when the guard-part is empty). Operationally it divides the right-hand side of a clause into two parts which are solved strictly in sequence. Before starting solving the body the whole guard must be solved. Literals in the guard and body are separated by commas. Operationally this means that the literals may be solved in parallel. The notion of derivation basically carries over from definite programs. However, the read-only annotations and commit operators impose certain restrictions on the selection of a subgoal in a derivation step. This is because some subgoals may be temporarily suspended. There are three reasons for this — either because (1) unification of a subgoal with a clause head cannot be performed without binding read-only
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variables or (2) the subgoal appears in the body of a guarded clause whose guard is not yet satisfied or (3) the subgoal is a non-ground test-predicate. To describe the basic derivation-step taking these restrictions into account the goal will be partitioned into groups of guards and bodies. To emphasize this the goal will be written as follows: ← G1 | B1 , . . . , Gi | Bi , . . . , Gn | Bn . where both Gj and Bj , (1 ≤ j ≤ n), are possibly empty conjunctions of atoms (in case of Gj containing only test-predicates). A single reduction of the goal then amounts to selecting some subgoal A such that either: (i ) A is a test-predicate in Gi or Bi (if Gi is empty) which is both ground and true. The new goal is obtained by removing A from the goal. (ii ) Gi is empty, A appears in Bi and is a user-defined predicate and there is a (renamed) guarded clause of the form: H ← Gm | Bm . such that A and H unify (with mgu θ) without binding any read-only variables. The new goal obtained is: (← G1 | B1 , . . . , Bi \ A , Gm | Bm , . . . , Gn | Bn )θ. where Bi \ A denotes the result of removing A from Bi . A successful derivation is one where the final goal is empty. Like SLD-resolution this scheme contains several nondeterministic choices — many subgoals may be selected and if the subgoal selected is user-defined there may be several guarded clauses which unify with it. In the latter case the commit operator has an effect similar to that of cut. In order to solve a subgoal several clauses are tried in parallel. However, as soon as the subgoal unifies with one of the clauses and succeeds in solving its guard, all other attempts to solve the subgoal are immediately surrendered. Thus, the commit operator behaves as a kind of symmetric cut. For instance, take Example 12.3 and the goal: ← merge([a, b], [c, d], X). This goal has many solutions in Prolog. In FCP there is only one solution to the goal. The result depends on what clauses the refutation commits to. Since each clause is required to contain exactly one commit operator no goal can have more than one solution. Thus, it is not possible to use append/3 to generate splittings of a list. At most one solution will be found. This is one of the main disadvantages of this class of languages. However, this is the price that has to be paid in order to be able to implement these languages efficiently. Note that it is vital to test for inequality in the guard of the last clause of the delete/3-program for this reason. If the test is moved to the body it may happen that goals fail because of committing to the third clause instead of the second.
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user Σ
dbms
user
Figure 12.2: Transaction system
The execution model given above is somewhat simplified since at each step only one subgoal is selected. As already mentioned, languages like FCP support ANDparallelism which means that several subgoals may be selected simultaneously. However, incorporating this extra dimension into the execution model above makes it rather complicated and we will therefore stick to the sequential version which simulates parallelism through coroutining. The chapter is concluded with an example of a CLP program that implements a simple database system with a fixed number of clients. Example 12.4 Consider an application involving a database transaction system. Such a system consists of some processes where customers (users) input transactions and a database management system (DBMS) performs the transactions using a database and outputs the results to the user. See Figure 12.2. To start up such a system of processes the following goal may be given (if we restrict ourselves to two users of the database system): ← user(tty1 , X), user(tty2 , Y ), merge(X?, Y ?, Z), dbms(Z?, [ ]). A formal definition of the user/2-process will not be provided. Informally the process user(ttyn , X) is assumed to behave as follows: (i ) It suspends until the arrival of a message from the terminal named ttyn ; (ii ) When a message M arrives it binds X to the pair [M |Msgs] where Msgs is a new variable. (For a description of all possible messages see Figure 12.3.); (iii ) Then the process suspends until M becomes ground; (iv ) When M becomes ground it prints a message on ttyn ; (v ) Finally it calls itself with user(ttyn , Msgs). Thus, the two user/2-processes generate two (infinite) streams of transactions which are merged nondeterministically by the merge/3-process and the resulting stream is processed (and grounded) by the dbms/2-system. For the sake of simplicity, assume that the database consists of a list of pairs of the form item(key, value) where key is a unique identifier and value is the data associated with the key. Three different transactions are to be considered — a pair may be (1) added to the database, (2) deleted from the database, and (3) retrieved from the database. On the top level the database management system may be organized as follows (the first clause is not needed in this version of the program but is added as a hint to exercise 12.3):
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Transactions add(key, value, Reply)
del(key, Reply)
in(key, Reply)
Description This message represents an external request to add a new item of the form item(key, value) to the database. The first two arguments are ground and Reply a variable which eventually is grounded by the DBMS. This message represents an external request to delete an item of the form item(key, ) from the database. The first argument is ground and Reply a variable which eventually is grounded by the DBMS. This message represents an external request to retrieve the value stored in the item of the form item(key, ) from the database. The first argument is ground and Reply a variable which eventually is grounded by the DBMS.
Figure 12.3: Description of database transactions
dbms([kill|N xt], Db). dbms([in(Key, Reply)|N xt], Db) ← retrieve(Key, Db, Reply), dbms(N xt?, Db). dbms([add(Key , V al, Reply)|N xt], Db) ← insert(Key, V al, Db?, N ewDb, Reply), dbms(N xt?, N ewDb?). dbms([del(Key, Reply)|N xt], Db) ← delete(Key, Db?, N ewDb, Reply), dbms(N xt?, N ewDb?). If the first transaction appearing in the stream is a request to retrieve information from the database, dbms/2 invokes the procedure retrieve/3. The first argument is the key sought for, the second argument is the current database and the third argument is the value associated with the key (or not found if the key does not appear in the database). retrieve(Key , [ ], not found ). retrieve(Key , [item(Key, X)|Db], X). retrieve(Key , [item(K, Y )|Db], X) ← Key 6= K | retrieve(Key, Db, X). If the first transaction is a request to add a new key/value-pair to the database the data is stored in the database by means of the predicate insert/5. The first and second arguments are the key and the associated value, the third argument is the current database, the fourth argument is the new database after adding the pair and the final argument returns a reply to the user (since the operation always succeeds the reply is always done).
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insert(Key, X, [ ], [item(Key, X)], done). insert(Key, X, [item(Key, Y )|Db], [item(Key, X)|Db], done). insert(Key, X, [item(K, Y )|Db], [item(K, Y )|N ewDb], Reply) ← Key 6= K | insert(Key, X, Db, N ewDb, Reply). Notice that insert/5 either adds the pair at the very end of the database or, if the key is already used in the database, replaces the old value associated with the key by the new value. If the latter is not wanted an error message may be returned instead by simple modifications of the second clause. The last transaction supported is the removal of information from the database. This is taken care of by the predicate delete/4. The first argument is the key of the pair to be removed from the database — the second argument. The third argument will be bound to the new database and the fourth argument records the result of the transaction (done if the key was found and not found otherwise): delete(Key, [ ], [ ], not found ). delete(Key, [item(Key, Y )|Db], Db, done). delete(Key, [item(K, Y )|Db], [item(K, Y )|N ewDb], Reply) ← Key 6= K | delete(Key, Db, N ewDb, Reply). We conclude the example by considering an outline of an execution trace of the goal: ← user(tty1 , X), user(tty2 , Y ), merge(X?, Y ?, Z), dbms(Z?, [ ]). Initially, all subgoals are suspended. Now assume that the first user/2-process binds X to [add(k10 , john, R)|X0 ]: ← . . . , merge([add(k10 , john, R)|X0 ], Y ?, Z), dbms(Z?, [ ]). Then merge/3 can be resumed binding Z to [add(k10 , john, R)|W1 ] and the new goal becomes: ← . . . , merge(X0 ?, Y ?, W1 ), dbms([add(k10 , john, R)|W1 ], [ ]). At this point dbms/2 is resumed reducing the goal to: ← . . . , merge(. . .), insert(k10 , john, [ ], D, R), dbms(W1 ?, D?). The call to insert/5 succeeds binding D to [item(k10 , john)] and R to done (the reply done is echoed on tty1 ): ← . . . , merge(X0 ?, Y ?, W1 ), dbms(W1 ?, [item(k10 , john)]). At this point both merge/3 and dbms/2 are suspended waiting for new messages from one of the terminals. Assume that the second user wants to know the value associated with the key k10 . Then Y is bound to [in(k10 , R)|Y2 ]: ← . . . , merge(X0 ?, [in(k10 , R)|Y2 ], W1 ), dbms(W1 ?, [item(k10 , john)]). Next W1 is bound to [in(k10 , R)|W3 ] and the goal is reduced to: ← . . . , merge(X0 ?, Y2 ?, W3 ), dbms([in(k10 , R)|W3 ], [item(k10 , john)]).
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Thereafter dbms/2 is resumed and unified with the second clause yielding the goal: ← . . . , merge(. . .), retrieve(k10 , [item(k10 , john)], R), dbms(W3 ?, . . .). The call to retrieve/3 succeeds with R bound to john. The answer is echoed on tty2 and the goal is reduced to: ← . . . , merge(X0 ?, Y2 ?, W3 ), dbms(W3 ?, [item(k10 , john)]). At this point the whole system is suspended until one of the users supplies another transaction. The example above illustrates one fundamental difference between sequential SLDresolution for definite programs and concurrent execution. In the former case computations are normally finite and the program computes relations. However, in the latter case, computations may be infinite and the meaning of the program is not so easily defined in terms of relations. For instance, when giving a goal: ← A1 , . . . , An we normally want this goal to succeed with some answer substitution. However, the goal in Example 12.4 does not terminate, yet the execution results in some useful output via side-effects (supplying transactions to the terminal and obtaining answers echoed on the screen). This fundamental difference makes more complicated to give a declarative semantics to concurrent logic programming languages like FCP.
Exercises 12.1 Write a concurrent logic program for checking if two binary trees have the same set of labels associated with the nodes of the tree. Note that the labels associated with corresponding nodes do not have to be the same. 12.2 Write a concurrent program for multiplying N × N -matrices of integers (for arbitrary N ’s). 12.3 Suggest a way of including a “kill”-process in Example 12.4. Such a process is initially suspended but should, when it is activated, terminate all other processes in the transaction system in a controlled way. 12.4 Write a concurrent program which takes as input a stream of letters (represented by constants) and replaces all occurrences of the sequence “aa” by “a” and all occurrences of “–” by the empty string. All other letters should appear as they stand in the input. 12.5 Give a solution to the producer-consumer problem with a bounded buffer.
Chapter 13 Logic Programs with Equality
As emphasized in the previous chapters, logic programs describe relations. Of course, since a function may be viewed as a special case of a relation it is also possible to define functions as relations using logic programs. However, in this case it is usually not clear whether the described relation is a function or not (cf. Section 5.2). Furthermore, this kind of description associates functions with predicate symbols, while it would be more desirable to have functions associated with functors. In this chapter we present a mechanism that allows us to incorporate such functional definitions into logic programming. The idea is to introduce a special binary . predicate symbol “=” — called the equality — which is to be interpreted as the identity relation on the domain of any interpretation of logic programs. The notion of equality thus makes it possible to restrict attention to interpretations where certain terms are identified. For instance the factorial function may be defined by the following (implicitly universally quantified) equations: . fac(0) = s(0). . fac(s(X)) = s(X) ∗ fac(X). . 0 ∗ X = 0. . s(X) ∗ Y = X ∗ Y + Y. . 0 + X = X. . s(X) + Y = s(X + Y ). In any model of these formulas the meanings of the terms fac(0) and s(0) are the same. The use of such equations may be exploited to extend the notion of unification. Consider a definite program: odd (s(0)). odd (s(s(X))) ← odd (X). The formula odd (fac(0)) certainly is true in the intended interpretation. However, 203
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since SLD-resolution is based on the notion of syntactic equality it is not powerful enough to produce an SLD-refutation from the definite goal: ← odd (fac(0)). In Section 13.3 we will see how definite programs with equality can be used to extend the notion of unification into so-called E-unification. However, before involving definite programs we study the meaning of equality axioms similar to those used above.
13.1
Equations and E-unification
. In what follows an equation will be a formula of the form s = t where s and t are terms from a given alphabet. This kind of unconditional equation may be extended to conditional ones. That is, formulas of the form: . F ⊃ (s = t) where s and t are terms and F is some formula possibly containing other predicate . symbols than “=”. In this chapter attention is restricted to unconditional equations. At a first glance the restriction to unconditional equations may seem to be a serious limitation, but from a theoretical point of view unconditional equations are sufficient to define any computable function (e.g. Rogers (1967)). Hence, the restriction is solely syntactic. . The intuition behind introducing the new predicate symbol “=”, is to identify terms which denote the same individual in the domain of discourse, regardless of the . values of their variables. Hence, for two terms s and t, the formula s = t is true in an interpretation = and valuation ϕ iff s and t have identical interpretations in = and ϕ . (that is, if ϕ= (s) = ϕ= (t)). An interpretation = is said to be a model of ∀(s = t) if . s = t is true in = under any valuation ϕ. This extends to sets E of equations: = is a model of a set E of equations iff = is a model of each equation in E. The concept of logical consequence carries over from Chapter 1 with the modifi. cation that the only interpretations that are considered are those that associate “ =” . with the identity relation. Hence, given a set of equations E, s = t is said to be a . . logical consequence of E (denoted E |= s = t) iff s = t is true in any model of E. One of the main objectives of any logic is to permit inference of new formulas from old ones using a system of rewrite rules. The inference rules in Figure 13.1 defines the relation between a set of equational hypohesis E and new derived equations. (The . . notation E ` s = t should be read “s = t is derived from E”.) The derivability relation induces an equivalence relation, ≡E , on the set of all terms, defined by s ≡E t . iff E ` s = t. The relation is called an equality theory. The inference rules just introduced were shown to be both sound and complete by Birkhoff (1935): Theorem 13.1 (Soundness and Completeness) . . E |= s = t iff E ` s = t iff s ≡E t The notion of E-unification is defined relative to the equality theory, ≡E , induced by E and `.
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Hypothesis: . . E ` s = t (if s = t ∈ E) Reflexivity: . E`s=s Symmetry: . E`s=t . E`t=s Transitivity: . . E`r=s E`s=t . E`r=t Stability: . E`s=t . E ` sθ = tθ Congruence: . . E ` s1 = t1 · · · E ` sn = tn . E ` f (s1 , . . . , sn ) = f (t1 , . . . , tn ) Figure 13.1: Inference rules for equality
Definition 13.2 (E-unifier) Two terms, s and t, are said to be E-unifiable if there exists some substitution θ such that sθ ≡E tθ. The substitution θ is called an E-unifier of s and t. Example 13.3 Let E be the following equalities defining addition of natural numbers: . sum(0, X) = X. . sum(s(X), Y ) = s(sum(X, Y )). Consider the problem of finding an E-unifier of the two terms sum(s(X), Y ) and s(s(0)). As formally shown in Figure 13.2, the two terms have at least one E-unifier — namely {X/0, Y /s(0)}. Note that in the case of the empty equality theory, ≡? relates every term only to itself. This implies that two terms, s and t, are ?-unifiable iff there is some substitution θ such that sθ is identical to tθ. Hence, the notion of E-unification encompasses “standard” unification as a special case.
13.2
More on E-unification
As observed above, standard unification as defined in Chapter 3 is a special case of E-unification for the degenerate case when E = ? . This suggests that it may be possible to generalize SLD-resolution into something more powerful by replacing
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. E ` sum(0, X) = X . . E ` sum(s(X), Y ) = s(sum(X, Y )) E ` sum(0, s(0)) = s(0) . . E ` sum(s(0), s(0)) = s(sum(0, s(0))) E ` s(sum(0, s(0))) = s(s(0)) . E ` sum(s(0), s(0)) = s(s(0)) . Figure 13.2: Proof of E ` sum(s(0), s(0)) = s(s(0))
standard unification by E-unification. Such an extension is discussed in the next section but first a number of questions are raised concerning the practical problems of E-unification. First of all, an E-unification algorithm must be provided. For the case when E = ? there are efficient unification algorithms available as discussed in Chapter 3. The algorithm given there has some nice properties — it always terminates and if the terms given as input to the algorithm are unifiable, it returns a most general unifier of the terms; otherwise it fails. For arbitrary sets of equations these properties are not carried over. For instance, E-unification is undecidable. That is, given an arbitrary set E of equations and two terms s and t, it is not in general possible to determine whether s ≡E t. In addition, the algorithm of Chapter 3 is complete in the sense that if s and t are unifiable, then any of their unifiers can be obtained by composing the output of the algorithm with some other substitution. This is because existence of a unifier implies the existence of a most general one. This is not true for arbitrary sets of equations. Instead a set of unifiers must be considered. Before resorting to an example, some preliminaries are needed to formulate this more precisely. A term t is said to subsume the term s iff there is a substitution σ such that tσ ≡E s. This is denoted by s E t. The relation can be extended to substitutions as follows — let V be a set of variables and σ, θ substitutions. Then θ subsumes σ relative to V (denoted σ E θ[V ]) iff Xσ E Xθ for all X ∈ V . If V is the set of all variables in s and t, then the set S of substitutions is a complete set of E-unifiers of s and t iff: • every θ ∈ S is an E-unifier of s and t; • for every E-unifier σ of s and t, there exists θ ∈ S such that σ E θ[V ]. For the case when E = ? the standard unification algorithm produces a complete set of E-unifiers. This set is either empty (if the terms are not unifiable) or consists of a single mgu. Unfortunately, for nonempty sets E of equations, complete sets of E-unifiers may be arbitrary large. In fact, there are cases when two terms only have an infinite complete set of E-unifiers. The following example shows two terms with two E-unifiers where the first Eunifier is not subsumed by the other and vice versa. Example 13.4 Consider the equations in Example 13.3 again. As shown in Figure 13.2 the substitution θ := {X/0, Y /s(0)} is an E-unifier of the terms sum(s(X), Y ) and s(s(0)). However, also σ := {X/s(0), Y /0} is a unifier of the terms. (The proof is
13.3 Logic Programs with Equality
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left as an exercise). It can be shown that neither θ E σ[{X, Y }] nor σ E θ[{X, Y }]. It can also be shown that any other E-unifier of the two terms is subsumed by one of these two substitutions. Thus, the set {θ, σ} constitutes a complete set of E-unifiers of the two terms. An E-unification algorithm is said to be sound if, for arbitrary terms s and t, its output is a set of E-unifiers of s and t. The algorithm is complete if the set in addition is a complete set of E-unifiers. Needless to say, it is desirable to have an E-unification algorithm which is at least sound and preferably complete. However, as already pointed out, there are sets of equations and pairs of terms which do not have finite sets of Eunifiers. For such cases we cannot find a complete E-unification algorithm. Thus, one must weaken the notion of completeness by saying that an algorithm is complete if it enumerates a complete set of E-unifiers (for arbitrary pairs of terms). Under this definition there are both sound and complete E-unification algorithms for arbitrary sets E of equations. Unfortunately they are of little practical interest because of their tendency to loop. Thus, instead of studying general-purpose algorithms, research has concentrated on trying to find algorithms for restricted classes of equations, much like research on logic programming started with the restricted form of definite programs. Standard unification is a trivial example where no equations whatsoever are allowed. The most well-known approach based on restricted forms of equations is called narrowing which, in many ways, resembles SLD-resolution. It has been shown to be both sound and complete for a nontrivial class of equational theories. Characterizing this class more exactly is outside the scope of this book. Unfortunately narrowing also suffers from termination problems. The reason is that the algorithm does not know when it has found a complete set of unifiers. It may of course happen that this set is infinite in which case there is no hope for termination whatsoever. But even if the set is finite, the algorithm often loops since it is not possible to say whether the set found so far is a complete set of E-unifiers. Hence, in practice one has to impose some sort of restrictions not only on the form of the equations but also on the terms to be E-unified. One simple case occurs when both terms are ground. In this case either ? or the singleton {} is a complete set of E-unifiers of the terms.
13.3
Logic Programs with Equality
In this section we review the integration of definite programs and equations. It turns out that the proof-theoretic and model-theoretic semantics of this language are natural extensions of the corresponding concepts for definite programs alone. But before describing the nature of these extensions the syntax of definite programs with equations is given. Thereafter weaknesses of definite programs alone are discussed to motivate the extensions. A definite program with equality is a pair P, E where: . • P is a finite set of definite clauses not containing the predicate symbol “=”; • E is a possibly infinite set of equations.
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One sometimes sees different extensions of this idea where E may contain e.g. con. ditional equations or where “=” may appear in the bodies of clauses in P . What is described below can also be generalized to such programs with some additional effort. Now, consider the following definite program P where the symbols have their natural intended interpretations: odd(1). odd(X + 2) ← odd(X). Although odd(2 + 1) is true in the intended model it is not a logical consequence of the program because the program has at least one model (for instance the least Herbrand model MP ) where odd(2 + 1) is false. It may thus be argued that the least Herbrand model is “incompatible” with the intended interpretation since the two terms 1 + 2 and 2 + 1 have distinct interpretations in MP — recall that any ground term denotes itself in any Herbrand interpretation. As pointed out above equations may be used to focus attention on certain models — namely those where some terms denote the same object. For instance, by adding to P the equation E: . 2+1=1+2 . (or more generally X + Y = Y + X) it is possible to exclude certain unwanted interpretations from being models of P and E. In particular, MP is no longer a model of both P and E. (In fact, no Herbrand interpretation of P is a model of P and E since the terms 1 + 2 and 2 + 1 denote distinct objects.) We recall that the model-theoretic semantics of definite programs without equality enjoys some attractive properties: To characterize the meaning of a program (i.e. its set of ground, atomic logical consequences) it is sufficient to consider the set of all Herbrand models. In fact, attention may be focused on a single least Herbrand model. Evidently, this is not applicable to definite programs with equality. However, there is a natural extension of these ideas: Instead of considering interpretations where the domain consists of ground terms one may consider interpretations where the domain consists of sets of equivalent ground terms. More precisely one may consider the quotient set of UP with respect to a congruence relation. Such a set will be called an E-universe. In what follows, it will be clear from the context what congruence relation is intended, and we will just write s to denote the equivalence class which contains s. By analogy to definite programs the E-base will be the set: {p(t1 , . . . , tn ) | t1 , . . . , tn ∈ E-universe and p/n is a predicate symbol} and an E-interpretation will be a subset of the E-base. The intuition behind an Einterpretation is as follows: (1) the meaning of a ground term t is the equivalence class . t and (2) if s and t are ground terms, then s = t is true in the interpretation iff s and t are members in the same equivalence class of the domain (i.e. if s = t). To characterize the set of all ground, atomic logical consequences of a program P, E we first define a set of E-interpretations which are models of E. Then we consider E-interpretations which are also models of P . The following theorem shows that it is reasonable to restrict attention to E-interpretations whose domain is UP /≡E (the set of all equivalence-classes of UP w.r.t. the relation ≡E ), since they characterize the set of all ground equations which are logical consequences of E:
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Theorem 13.5 Let E be a set of equations, s and t ground terms and = an Einterpretation whose domain is UP /≡E . Then: . = |= s = t iff iff iff
s=t s ≡E t . E |= s = t
Such E-interpretations are called canonical. Notice that if E = ? then s = {s} for any ground term s, and = reduces to a Herbrand interpretation (except that the domain consists of singleton sets of ground terms). Example 13.6 Consider the following set E of equations: . father (sally) = robert. . father (bruce) = adam. . father (simon) = robert. Then UP /≡E contains elements such as: robert adam sally bruce
= = = =
{robert, father (sally), father(simon)} {adam, father (bruce)} {sally} {bruce}
Most of the results from Chapter 2 can be carried over to canonical E-interpretations. For instance (see Jaffar, Lassez and Maher (1986) or (1984) for details): • if P, E has a model then it also has a canonical E-model; • the intersection of all canonical E-models of P, E is a canonical E-model; • there is a least canonical E-model (denoted by MP,E ). Moreover, MP,E characterizes the set of all ground, atomic logical consequences of P, E. In what follows let p(t1 , . . . , tn ) be an abbreviation of p(t1 , . . . , tn ). If t1 , . . . , tn are ground terms, then: P, E |= p(t1 , . . . , tn ) iff
p(t1 , . . . , tn ) ∈ MP,E
An alternative characterization of this set can be given by a fixed point-operator similar to the TP -operator for definite programs. The operator — denoted by TP,E — is defined as follows: TP,E (x) := {A | A ← A1 , . . . , An ∈ ground(P ) ∧ A1 , . . . , An ∈ x} Jaffar, Lassez and Maher (1984) showed that MP,E = TP,E ↑ ω.
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Example 13.7 Let P, E be the program: proud(father (X)) ← newborn(X). newborn(sally). newborn(bruce). . father (sally) = robert. . father (bruce) = adam. . father (simon) = robert. In this case: MP,E = {proud(robert), proud(adam), newborn(sally), newborn(bruce)}
As observed above the model-theoretic semantics of definite programs with equality is a generalization of the model-theoretic semantics of definite programs. This is not particularly strange since a definite program also has a set of equations, albeit empty. One might expect that a similar situation would crop up for the proof-theoretic semantics and, indeed, it does. In principle, the only modification which is needed to SLD-resolution is to replace ordinary unification by E-unification. In what follows we presuppose the existence of a complete E-unification algorithm. However, we do not spell out how it works. The following informal description describes the principles of the proof-theory. Let P be a definite program, E a set of equations and G the definite goal: ← A1 , . . . , Am−1 , Am , Am+1 , . . . , An Now assume that C is the (renamed) program clause: B0 ← B1 , . . . , Bj
(j ≥ 0)
and that Am and B0 have a nonempty, complete set of E-unifiers Θ. Then G and C resolve into the new goal: ← (A1 , . . . , Am−1 , B1 , . . . , Bj , Am+1 , . . . , An )θ if θ ∈ Θ. To avoid confusing this with ordinary SLD-resolution it will be called the SLDEresolution principle. The notion of SLDE-derivation, refutation etc. are carried over from Chapter 3. SLDE-resolution introduces one extra level of nondeterminism — since two atoms may have several E-unifiers none of which subsume the others, it may happen that a given computation rule, a goal and a clause with a head that E-unifies with the selected subgoal, result in several new goals. This was not the case for SLD-resolution since the existence of a unique mgu allowed only one new goal to be derived. Example 13.8 Consider again the following definite program and equations:
13.3 Logic Programs with Equality
211
← proud(robert).
,@@ , , @
← newborn(sally). ← newborn(simon).
2 Figure 13.3: SLDE-tree for the goal ← proud(robert)
proud(father (X)) ← newborn(X). newborn(sally). newborn(bruce). . father (sally) = robert. . father (bruce) = adam. . father (simon) = robert. Let G0 be the goal: ← proud(robert). Since {{X0 /sally}, {X0/simon}} is a complete set of E-unifiers of proud(robert) and proud(father (X0 )), G1 is either of the form: ← newborn(sally). which results in a refutation; or of the form: ← newborn(simon). which fails since simon E-unifies neither with sally nor with bruce. By analogy to SLD-resolution all (complete) SLDE-derivations under a given computation rule may be depicted in a single SLDE-tree (cf. Figure 13.3). Notice in contrast to SLD-trees, that the root of the tree has two children despite the fact that the definition of proud/1 contains only one clause. Since the complete set of E-unifiers of two terms may be infinite, a node in the SLDE-tree may have infinitely many children — something which is not possible in ordinary SLD-trees. This, of course, may cause operational problems since a breadth-first traversal is, in general, not sufficient for finding all refutations in the SLDE-tree. However, soundness and completeness results similar to those for SLD-resolution can and have been proved also for SLDE-resolution described above.
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Chapter 13: Logic Programs with Equality
Exercises 13.1 Consider the equations in Example 13.3. Prove that {X/s(0), Y /0} is an Eunifier of sum(s(X), Y ) and s(s(0)). 13.2 Show that the inference rules in Figure 13.1 are sound. Try to prove that they are complete! 13.3 Consider the following equations: . append (nil , X) = X . append (cons(X, Y ), Z) = cons(X, append (Y, Z)) Prove that append (X, cons(b, nil)) and cons(a, cons(b, nil )) are E-unifiable. 13.4 Prove that two terms, s and t, are ?-unifiable iff there is some substitution θ such that sθ and tθ are syntactically identical.
Chapter 14 Constraint Logic Programming
The following program (which describes the property of being a list whose elements are sorted in ascending order) is intended to illustrate some shortcomings of SLDresolution as previously presented: sorted ([ ]). sorted ([X]). sorted ([X, Y |Xs]) ← X ≤ Y, sorted([Y |Xs]). Consider the query “Are there integers X, Y and Z, such that the list [X, Y, Z] is sorted?”. The query may be formalized as a goal: ← sorted ([X, Y, Z]).
(G0 )
SLD-resolution, as described in Chapter 3, attempts to construct a counter-example — a substitution θ such that P ∪ {∀(¬sorted ([X, Y, Z])θ)} is unsatisfiable. If such a counter-example can be constructed, then most Prolog systems present the answer as a set of variable bindings, e.g. {X/1, Y /2, Z/3}. This may be interpreted as follows: . . . ∀(X = 1 ∧ Y = 2 ∧ Z = 3 ⊃ sorted ([X, Y, Z])) Now the goal G0 may be reduced to: ← X ≤ Y, sorted ([Y, Z]).
(G1 )
In a Prolog implementation this would lead to a run-time error since arithmetic tests can be made only if the arguments are instantiated. This could easily be repaired by imagining an infinite set of facts of the form 0 ≤ 0, 0 ≤ 1, . . . Unfortunately this would lead to an infinite number of answers to the original query. Assume instead that the interpreter is clever enough to realize that X ≤ Y has at least one solution. Then the second subgoal of G1 may be selected instead, in which case G1 is reduced to: ← X ≤ Y, Y ≤ Z, sorted([Z]). 213
(G2 )
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Chapter 14: Constraint Logic Programming
Under the assumption that the interpreter is intelligent enough to realize that ∃(X ≤ Y ∧ Y ≤ Z) is satisfiable, the final recursive call can be eliminated: ← X ≤ Y, Y ≤ Z.
(G3 )
If ∃(X ≤ Y ∧ Y ≤ Z) is satisfiable, then G3 — i.e. ∀¬(X ≤ Y ∧ Y ≤ Z) — is unsatisfiable. Thus, G0 has a refutation. Moreover, X ≤ Y, Y ≤ Z may be viewed as an answer to G0 : ∀(X ≤ Y ∧ Y ≤ Z ⊃ sorted ([X, Y, Z])) The example illustrates two desirable extensions of logic programming. First the use of dedicated predicate symbols (in our case ≤) whose semantics are built into the interpreter rather than defined by the user. Second the extension of the notion of an answer not just to include equations. These extensions have spawned a number of new logic programming languages incorporating various built-in domains, operations and relations. They are all instances of a more general scheme known as constraint logic programming (CLP). This chapter surveys the theoretical foundations of constraint logic programming languages and discusses some specific instances of the scheme.
14.1
Logic Programming with Constraints
In the previous chapter logic programming was extended with a dedicated predicate . symbol =, always interpreted as the identity relation. Constraint logic programming languages generalize this idea by allowing also other dedicated interpreted predicates, function symbols and constants (in addition to the uninterpreted symbols). Hence, a CLP language CLP (D) is parameterized by an interpretation D of certain symbols in the language. Definition 14.1 (Constraint logic program) A constraint logic program is a finite set of clauses: A0 ← C1 , . . . , Cm , A1 , . . . , An
(m, n ≥ 0)
where C1 , . . . , Cm are formulas built from the interpreted alphabet (including variables, quantifiers and logical connectives) and A0 , . . . , An are atoms with uninterpreted predicate symbols. The formulas C1 , . . . , Cm are called constraints. In all our examples, a constraint will be an atomic formula with an interpreted predicate symbols. All CLP languages are . assumed to contain the predicate = which is always interpreted as identity. Let Z be an interpretation of numerals such as 0, 1, . . ., function symbols such . as +, −, . . . and predicates such as =, 0, W is X mod Y, gcd(Y, W, Z). 6.2 For instance: grandchild(X, Z) ← parent(Y, X), parent(Z, Y ). sister(X, Y ) ← f emale(X), parent(Z, X), parent(Z, Y ), X 6= Y. brother(X, Y ) ← male(X), parent(Z, X), parent(Z, Y ), X 6= Y. etc. 6.3 Hint: (1) Colours should be assigned to countries. Hence, represent the countries by variables. (2) Describe the map in the goal by saying which countries should be assigned different colours. 6.4 For instance: and(1, 1, 1). and(0, 1, 0). and(1, 0, 0). and(0, 0, 0). inv(1, 0). inv(0, 1). circuit1(X, Y, Z) ← and(X, Y, W ), inv(W, Z). circuit2(X, Y, Z, V, W ) ← and(X, Y, A), and(Z, V, B), and(A, B, C), inv(C, W ). 6.5 For instance: p(X, Y ) ← husband(K, X), wif e(K, Y ). q(X) ← parent(X, Y ). q(X) ← income(X, Y ), Y ≥ 20000.
Appendix C: Answers to Selected Exercises
257
6.6 For instance: πX,Y (Q(Y, X)) ∪ πX,Y (Q(X, Z) 1 R(Z, Y )) 6.7 For instance: compose(X, Z) ← r1 (X, Y ), r2 (Y, Z). 6.9 Take the transitive closure of the parent/2-relation. 6.11 For instance: ingredients(tea, needs(water, needs(tea bag, nil))). ingredients(boiled egg, needs(water, needs(egg, nil))). available(water). available(tea bag). can cook(X) ← ingredients(X, Ingr), all available(Ingr). all available(nil). all available(needs(X, Y )) ← available(X), all available(Y ). needs ingredient(X, Y ) ← ingredients(X, Ingr), among(Y, Ingr). among(X, needs(X, Y )). among(X, needs(Y, Z)) ← among(X, Z). 7.1 Alternative list notation:
.(a, .(b, [ ])) .(a, b) .(a, .(.(b, .(c, [ ])), .(d, [ ]))) .(a, .(b, X))
.(a, .(b, .(c, [ ]))) .(a, .(b, [ ])) .([ ], [ ]) .(a, .(b, .(c, [ ])))
7.4 For instance: length([ ], 0). length([X|Y ], N ) ← length(Y, M ), N is M + 1. 7.5 For instance: lshift ([X|YZ ], YZX ) ← append (YZ , [X], YZX ).
258
Appendix C: Answers to Selected Exercises
7.9 For instance: sublist (X, Y ) ← prefix (X, Y ) sublist (X, [Y |Z]) ← sublist (X, Z). 7.12 For instance: msort([ ], [ ]). msort([X], [X]). msort(X, Y ) ← split(X, Split1 , Split2 ), msort(Split1 , Sorted1 ), msort(Split2 , Sorted2 ), merge(Sorted1 , Sorted2 , Y ). split([X], [], [X]). split([X, Y |Z], [X|V ], [Y |W ]) ← split(Z, V, W ). merge([ ], [ ], [ ]). merge([X|A], [Y |B], [X|C]) ← X < Y, merge(A, [Y |B], C). merge([X|A], [Y |B], [Y |C]) ← X ≥ Y, merge([X|A], B, C). 7.13 For instance: edge(1, 2, b). edge(2, 2, a). edge(2, 3, a). edge(3, 2, b). final(3). accept(State, [ ]) ← final(State). accept(State, [X|Y ]) ← edge(State, N ewState, X), accept(N ewState, Y ). 7.17 For instance: palindrome(X) ← diff palin(X − [ ]). diff palin(X − X). diff palin([X|Y ] − Y ). diff palin([X|Y ] − Z) ← diff palin(Y − [X|Z]).
Appendix C: Answers to Selected Exercises
259
7.20 Hint: Represent the empty binary tree by the constant empty and the nonempty tree by node(X, Left, Right) where X is the label and Left and Right the two subtrees of the node. 8.3 The following program provides a starting point (the program finds all refutations but it does not terminate): prove(Goal) ← int(Depth), dfid(Goal, Depth, 0). dfid (true, Depth, Depth). dfid ((X, Y ), Depth, N ewDepth) ← dfid (X, Depth, T mpDepth), dfid (Y, T mpDepth, N ewDepth). dfid (X, s(Depth), N ewDepth) ← clause(X, Y ), dfid (Y, Depth, N ewDepth). int(s(0)). int(s(X)) ← int(X). 8.4 Hint: For instance, the fourth rule may be defined as follows: d(X + Y, Dx + Dy) ← d(X, Dx), d(Y, Dy). 10.3 Definite clause grammar: bleat → [b], aaa. aaa → [a]. aaa → [a], aaa. Prolog program: bleat(X0, X2 ) ← connects(X0 , b, X1 ), aaa(X1 , X2 ). aaa(X0 , X1 ) ← connects(X0 , a, X1 ). aaa(X0 , X2 ) ← connects(X0 , a, X1 ), aaa(X1 , X2 ). connects([X|Y ], X, Y ). A refutation is obtained, for instance, by giving the goal ← bleat([b, a, a], []). 10.4 The DCG describes a language consisting only of the empty string. However, at the same time it defines the “concatenation”-relation among lists. That is, the nonterminal x([a, b], [c, d], X) not only derives the empty string but also binds X to [a, b, c, d]. 12.1 The definition of append/3 and member/2 is left to the reader:
260
Appendix C: Answers to Selected Exercises eq(T 1, T 2) ← nodes(T 1, N 1), nodes(T 2, N 2), equal(N 1?, N 2?). nodes(empty, [ ]). nodes(tree(X, T 1, T 2), [X|N ]) ← nodes(T 1, N 1), nodes(T 2, N 2), append(N 1?, N 2?, N ). equal(X, Y ) ← subset(X, Y ), subset(Y, X). subset([ ], X). subset([X|Y ], Z) ← member(X, Z), subset(Y ?, Z).
12.2 Hint: write a program which transposes the second matrix and then computes all inner products. 13.3 Hint: The overall structure of the proof is as follows: . E ` app(nil, X) = X . . . E ` app(c(X, Y ), Z) = c(X, app(Y, Z)) E ` a = a E` = . . E` = E` = . E` = 14.3 The following program with real-valued or rational constraints can be used to answer e.g. the goal ← jugs([M, 1 − M, 0], Res, N ). jugs([A, B, C], [A, B, C], N ) ← . . N = 0, A + B + C = 1. jugs([A, B, C], Res, N ) ← N > 0, jugs([0.6 ∗ A + 0.2 ∗ B, 0.7 ∗ B + 0.4 ∗ A, C + 0.1 ∗ B], Res, N − 1). 15.1 The transformed program looks as follows: expr(X, Z) ← call expr(X, Z), expr(X, [+|Y ]), expr(Y, Z). expr([id|Y ], Y ) ← call expr([id|Y ], Y ). call expr(X, [+|Y ]) ← call expr(X, Z). call expr(Y, Z) ← call expr(X, Z), expr(X, [+|Y ]). Adding call expr([id, +, id], X) to the program yields the semi-naive iteration:
Appendix C: Answers to Selected Exercises ∆x0 ∆x1 ∆x2 ∆x3 ∆x4
= = = = =
{call expr([id, +, id], A)} {expr([id, +, id], [+, id]), call expr([id, +, id], [+|A])} {call expr([id], A), call expr([id], [+|A])} {expr([id], [ ])} {expr([id, +, id], [ ])}
261
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